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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIn genome-wide association (GWA) studies the aim is to test for association between genetic markers and a phenotype. A large number of markers are tested, and it is important to control the overall Type I error rate. Our focus is on controlling the familywise error rate (FWER). Multiple testing correction methods may achieve this goal by estimating a local significance level for the individual tests. In this work we present a new method, {\\em the order $k$ FWER-approximation method}, for finding a local significance level in multiple hypothesis testing for correlated common variants, as is often observed in GWA studies.\n\nAssume that we have collected independent individual observations in a case--control, cohort or cross-sectional study. The phenotype of interest can be continuous or discrete. We consider biallelic genetic markers, giving three possible genotypes. For each genetic marker we specify a hypothesis situation, where the null hypothesis is of the type ``no association between the phenotype and genetic marker'' and we have a two sided alternative. We will model the data using a generalized linear regression model (GLM) with phenotype as response (outcome), genotype as the independent variable of interest (exposure), and possibly non-genetical, referred to as environmental, independent covariates (not of interest) in the model. In epidemiological studies, a confounder is a common factor which is associated with both the exposure and outcome. In GWA studies, population substructure may be associated with both the exposure (genotype) and outcome (phenotype) and therefore may be a confounding factor and need to be adjusted for in the analysis. Population stratification can be adjusted for by including principal components of the genotype covariance matrix of the individuals as covariates in the model \\citep{Price2006}. As test statistics for the multiple hypothesis problem we use the score test statistics to evaluate the genotype contribution to the model for each genetic marker separately. It is known that the vector of separate score test statistics asymptotically follows a multivariate normal distribution with a covariance matrix that can be estimated using key features of the fitted GLM model and the genetic markers \\citep{Schaid2002, SeamanMyhsok2005}. This has also been a key ingredient in the work of \\cite{ConneelyBoehnke2007}.\n\nFurther, we show that for the special case when no environmental covariates are present or when environmental and genetic covariates are observed to be independent, the estimated correlation matrix between score test statistics can be approximated by the estimated correlation matrix between the genetic markers.\n\nIn a multiple testing situation with $m$ tests the familywise error rate can be controlled at level $\\alpha$ by specifying a local $p$-value cut-off, $\\alpha_{\\text{loc}}$, to be used for all the $m$ hypothesis tests. Inspired from the work of \\cite{MoskvinaSchmidt2008} and \\cite{DickhausStange2013} we will use an approximation to the $m$-dimensional asymptotic simultaneous multivariate normal distribution of the score test statistics vector to estimate $\\alpha_{\\text{loc}}$. The $\\alpha_{\\text{loc}}$ estimate can be used to define an effective number of independent tests, and our FWER-approximation can be used to compute FWER-adjusted $p$-values.\n\nThe order $k$ FWER-approximation method is more powerful than the \\v Sid\\'ak  method (which assumes that the score test statistics are independent across markers) and the Bonferroni method (which is valid for all dependence structures between the score test statistics). Further, it is more efficient and more widely applicable than the method of \\cite{ConneelyBoehnke2007}. In Section \\ref{sec:discuss} we will see that the method of \\cite{ConneelyBoehnke2007}  is built on numerical integration in $m$ dimensions and is computationally intensive. \n\nThe Westfall--Young permutation procedure is known to have asymptotically optimal power for a broad class of problems, including block-dependent and sparse dependence structure \\citep{MeinshausenMaathuisBuhlman2011}. However, this method is computer intensive and to have a valid permutation test, the assumption of exchangeability needs to be satisfied \\citep{Commenges2003}. This assumption is in general not satisfied when environmental covariates are present in the model.\n\nWe will use two genetic data sets presented by \\cite{TOP1}, \\cite{TOP2}, \\cite{VO2max1} and \\cite{VO2max2} to illustrate our method applied to real data. \n\nThe paper is organized as follows. In Section \\ref{sec:background} we present statistical background on the score test, and derive expressions for the score test covariance matrix, which is of importance for the subsequent work. Our proposed method is outlined and presented in detail in Section \\ref{sec:multtest}, together with characteristics of our method. In Section \\ref{sec:invest} real data and an artificial correlation structure are used to evaluate our proposed model and compare to other methods. Finally, we discuss and conclude in Sections~\\ref{sec:discuss} and~6.\n\n\\section{Statistical background}\\label{sec:background}\n\nIn this section, we present notation and details on the score test in generalized linear models. \n\n\\subsection{Notation and data} \\label{sec:notation}\n\nWe assume that data -- phenotype, $m$ genetic covariates and $d$ environmental covariates -- from $n$ independent individuals are available in a case--control, cohort or cross-sectional study. Let $\\bm Y$ be an $n$-dimensional vector having the phenotype $Y_i$ of individual $i$ as its $i$th entry, $i=1$, \\ldots, $n$. Let $\\xe$ be an $n\\times d$ matrix having environmental covariates (the first one being 1 to allow for an intercept in the model presented below) for individual $i$ as its $i$th row, and let $\\xg$ be an $n\\times m$ matrix having genetic covariates, or genotypes, for individual~$i$ as its $i$th row, each column corresponding to a genetic marker.\n\nWe assume that the genetic data are from common variant biallelic genetic markers with alleles $a$ and $A$, where $A$ is the minor allele. We will use the additive coding 0, 1, 2 for the genotypes $aa$, $aA$, and $AA$, respectively, in the genetic covariate matrix $\\xg$, but other coding schemes are also possible. We denote the total design matrix $\\x=(\\xe\\ \\xg)$, which has the total covariate vector for individual $i$ as its $i$th row.\n\n\\subsection{Testing statistical hypotheses with the score test}\\label{sec:score}\n\nWe assume that the relationship between the phenotype $\\y$ and covariates $\\x$ can be modelled by a generalized linear model (GLM) \\citep{glm} with an $n$-dimensional vector $\\bm\\eta = \\xe\\be + \\xg\\bg = \\x\\bm\\beta$ of linear predictors, where $\\bm\\beta=(\\be^T\\ \\bg^T)^T$ is a $d+m$-dimensional parameter vector. Let $\\eta_i$ be the $i$th entry of $\\bm\\eta$, and let $\\bm\\mu$ be the $n$-dimensional vector having $\\mu_i=EY_i$ as its $i$th entry. We assume that the link function $g$ defined by $\\eta_i=g(\\mu_i)$ of the GLM is canonical, which implies that the log likelihood for individual~$i$ is $l_i=(Y_i\\eta_i-b(\\eta_i))\/\\phi_i+c(Y_i,\\phi_i)$, where $b$ and $c$ are functions defining the exponential family of the phenotypes and $\\phi_i$ the dispersion parameter. In our context $\\phi_i=\\phi$ will be equal for all observations. In general, $\\mu_i=b'(\\eta_i)$ and $\\var\\y[i]=\\sigma^2_i=\\phi b''(\\eta_i)$. For $Y_i$ normally distributed, this reduces to $\\sigma_i^2=\\sigma^2=\\phi$, and for $Y_i$ Bernoulli distributed,  $\\sigma_i^2=\\mu_i(1-\\mu_i)$ with $\\phi=1$.\n\nThe full $d+m$-dimensional score vector $\\sum_{i=1}^n\\nabla_{\\!\\bm\\beta}l_i$ can then be calculated to be\n\\[\n \\bm U=\\frac1\\phi\\x^T(\\y-\\bm\\mu),\n\\]\nwhich is asymptotically normal with mean $\\bm0$ and covariance matrix \n\\[\n V=\\frac1{\\phi^2}\\x^T\\Lambda\\x,\n\\]\nwhere $\\Lambda$ is the diagonal matrix having $\\sigma_i^2$ as its $i$th entry.\n\nPartition $\\bm U$ into its environmental and genetic components, $\\bm U^T=(\\bm U_\\text e^T\\ \\bm U_\\text g^T)$. Since $\\be$ are nuisance parameters and unknown, they are estimated by their maximum likelihood estimates under the null hypothesis of $\\bg=\\bm0$. In effect, $\\bm\\mu$ is to be replaced by $\\hat{\\bm\\mu}_\\text e$, the fitted values in a model with only environmental covariates $\\xe$ present, giving the statistic\n\\begin{equation}\n \\bm U_{\\text g\\mid\\text e}=\\frac1\\phi\\xg^T(\\y-\\hat{\\bm\\mu}_\\text e).\\label{u}\n\\end{equation}\nThen $\\bm U_{\\text g\\mid\\text e}$ has the conditional distribution of $\\bm U_\\text g$ given $\\bm U_\\text e=\\bm 0$, which is asymptotically normal with mean $\\bm 0$ and covariance matrix\n\\begin{equation}\n V_{\\text g\\mid\\text e}^{\\vphantom1}=V_\\text{gg}^{\\vphantom1}-V_\\text{ge}^{\\vphantom1}V_\\text{ee}^{-1}V_\\text{eg}^{\\vphantom1}\n =\\frac{1}{\\phi^2}\\xg^T\\big(\\Lambda-\\Lambda    \\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T \\Lambda\\big)\\xg,\\label{var}\n\\end{equation}\nwhere $V_\\text{ee}$, $V_\\text{eg}$, $V_\\text{ge}$ and $V_\\text{gg}$ are the upper left $d\\times d$, upper right $d\\times m$, lower left $m\\times d$ and lower right $m\\times m$ submatrices of $V$, respectively \\citep[see][]{Smyth2003}.\n\nThe score test statistic $\\bm U_{\\text g\\mid\\text e}^TV_{\\text g\\mid\\text e}^{-1}\\bm U_{\\text g\\mid\\text e}$ with $\\bg=\\bm 0$ is asymptotically $\\chi^2$ distributed with $m$ degrees of freedom when the complete null hypothesis $\\bg=\\bm0$ is true \\citep[see][]{Smyth2003}. However, our interest lies not in the complete null hypothesis, but in the $m$ individual hypotheses $\\bg[j]=0$ for each component $\\bg[j]$ of $\\bg$, \\ $1\\leq j\\leq m$, against two-sided alternatives. We consider the standardized components of $\\bm U_{\\text g\\mid\\text e}$,\n\\begin{equation}\n T_j=\\frac{\\bm U_{\\text g\\mid\\text e\\,j}}{\\sqrt{V_{\\text g\\mid\\text e\\,jj}}}, \\label{eq:Tk}\n\\end{equation}\nwhere $\\bm U_{\\text g\\mid\\text e\\,j}$ denotes the $j$th entry of  $\\bm U_{\\text g\\mid\\text e}$ and $V_{\\text g\\mid\\text e\\,jk}$ the $jk$ entry of $V_{\\text g\\mid\\text e}$. Under the null hypothesis $H_j\\colon \\bg[j]=0$, \\ $T_j$ is asymptotically standard normally distributed, and $H_j$ will be rejected for large values of $\\lvert T_j\\rvert$. Under the complete null hypothesis, $\\bg=\\bm0$, the vector $\\bm T=(T_1,T_2,\\ldots,T_m)$ is asymptotically multivariate standard normally distributed with covariance matrix $R$, having\n\\begin{equation}\n \\cov(T_j,T_k)=\\frac{V_{\\text g\\mid\\text e\\,jk}}{\\sqrt{V_{\\text g\\mid\\text e\\,jj}V_{\\text g\\mid\\text e\\,kk}}}, \\label{eq:R}\n\\end{equation}\nas its $jk$ entry, all evaluated at $\\bg=\\bm0$. Note that the dispersion parameter $\\phi$ is cancelled from $\\bm T$ and the covariances. However, the $\\sigma_i^2$ of $\\Lambda$ will have to be estimated.\n\n\\subsection{Special cases} \\label{sec:corr}\n\nWe will now look at $\\bm U_{\\text g\\mid\\text e}$ and  $V_{\\text g\\mid\\text e}$ for some special cases.\n\n\\subsubsection{No environmental covariates}\\label{noenvcov}\n\nIf no evironmental covariates except the intercept are present in the GLM, then $\\xe=\\bm1$, the $n$-dimensional vector having all entries equal to 1, and $\\Lambda=\\sigma^2I$ under the null hypothesis, where $I$ is the $n\\times n$ identity matrix. Then\n\\[\n U_{\\text g\\mid\\text e}=\\frac1\\phi\\xg^T\\Big(I-\\frac{1}{n}\\bm1\\bm1^T\\Big)\\bm Y\\qquad\\text{and}\\qquad V_{\\text g\\mid\\text e}=\\frac{\\sigma^2}{\\phi^2}\\xg^T\\Big(I-\\frac{1}{n}\\bm1\\bm1^T\\Big)\\xg,\n\\]\nso that\n\\begin{equation}\nT_j=\\frac{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm Y}{\\sigma\\sqrt{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_j}},\\quad\\cov(T_j,T_k)=\\frac{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_k}{\\sqrt{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_j}\\sqrt{\\bm x_k^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_k}},\\label{Tk-noenv}\n\\end{equation}\nwhere $\\bm x_j$ is the $j$th column of $\\xg$, \\ $1\\leq j\\leq m$, \\ $1\\leq k\\leq m$. So $T_j$, the score test statistic for testing $\\bg[j]=0$, is $\\sqrt n$ times the Pearson correlation between $\\bm x_j$ and $\\bm Y$ when $\\sigma^2=\\var Y_i$ is replaced by the estimate $\\bm Y^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm Y\/n$, and $\\cov(T_j,T_k)$ is the sample correlation between $\\bm x_j$ and $\\bm x_k$. Thus, for a GLM without adjustment for environmental covariates, the correlation between the score test statistics can be estimated by estimating the genotype correlation. The genotype correlation estimates twice the composite linkage disequilibrium if the genotypes are coded 0, 1,~2 \\citep{Weir2008}.\n\n\\subsubsection{Uncorrelated environmental and genetic covariates}\\label{uncorrenvgen}\n\nTwo $n$-dimensional vectors $\\bm X_1$ and $\\bm X_2$ of observations have zero Pearson correlation if their centered observations are orthogonal,\n\\[\n 0=(\\bm X_1-\\bar X_1\\bm1)^T(\\bm X_2-\\bar X_2\\bm1)=\\bm X_1^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)\\bm X_2.\n\\]\nIf $X_1$ and $X_2$ are two matrices, then near zero Pearson correlation of each combination of a column of $X_1$ and a column of $X_2$ can be written compactly as\n\\begin{equation}\n X_1^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)X_2\\approx\\bm0,\\qquad\\text{or}\\qquad\n   X_1^TX_2\\approx\\frac1nX_1^T\\bm1\\bm1^TX_2.\\label{centeredorth}\n\\end{equation}\n\nIf we consider genetic and environmental covariates to be random variables, and all pairs of an environmental and a genetic covariate to be independent, we would expect~\\eqref{centeredorth} to  hold for all $X_1$ having columns that are functions of genetic covariates and $X_2$ having columns that are functions of environmental covariates. In particular, we consider $X_1=\\xg$ and $X_2=\\Lambda\\xe$. Since $\\Lambda$ is a function of environmental covariates only under the null hypothesis, so is $X_2$. By~\\eqref{centeredorth}, $\\xg^T\\Lambda\\xe\\approx\\frac1n\\xg^T\\bm1\\bm1^T\\Lambda\\xe$, Then, from~\\eqref{var},\n\\begin{align*}\n \\phi^2V_{\\text g\\mid\\text e}\n  &\\approx\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}\\xg^T\\bm1\\bm1^T\\Lambda\\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T \\Lambda\\bm1\\bm1^T\\xg\\\\\n  &=\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}\\xg^T\\bm1\\bm1^T\\Lambda^{1\/2}H\\Lambda^{1\/2}\\bm1\\bm1^T\\xg,\n\\end{align*}\nwhere $H=\\Lambda^{1\/2}\\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T\\Lambda^{1\/2}$ will project onto the column space of $\\Lambda^{1\/2}\\xe$. Since $\\bm1$ is a column (the intercept) of $\\xe$, \\ $\\Lambda^{1\/2}\\bm1$ is in the column space of $\\Lambda^{1\/2}\\xe$, so that $H\\Lambda^{1\/2}\\bm1=\\Lambda^{1\/2}\\bm1$, and\n\\[\n \\phi^2V_{\\text g\\mid\\text e}\n  \\approx\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}(\\tr\\Lambda)\\xg^T\\bm1\\bm1^T\\xg.\n\\]\n\nWe now turn to the term $\\xg^T\\Lambda\\xg$. Its $(j,k)$ entry is $\\bm X_1^T\\Lambda\\bm1$, where $\\bm X_1$ is the vector consisting of the entry-wise products of the $j$th and the $k$th column of $\\xg$. Letting $\\bm X_2=\\Lambda\\bm1$, by~\\eqref{centeredorth}, independence of environmental and genetic covariates yields $\\bm X_1^T\\Lambda\\bm1\\approx\\frac1n\\bm X_1^T\\bm1\\bm1^T\\Lambda\\bm1=\\frac1n(\\tr\\Lambda)\\bm X_1^T\\bm1$, which is the $(j,k)$ entry of $\\frac1n(\\tr\\Lambda)\\xg^T\\xg$. Thus $\\xg^T\\Lambda\\xg\\approx\\frac1n(\\tr\\Lambda)\\xg^T\\xg$, and we have\n\\[\n V_{\\text g\\mid\\text e} \\approx\\frac{\\tr\\Lambda}{n\\phi^2}\\xg^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)\\xg,\n\\]\nwhich is the same expression as in the case of no environmental covariates with the exception that the common variance $\\sigma^2$ of the responses is replaced by their average variance $\\tr\\Lambda\/n=\\frac1n\\sum_{i=1}^n\\sigma_i^2$, where the $\\sigma_i^2$ are defined by the environmental covariates. The conclusion is that, if environmental and genetic covariates are uncorrelated, correlations of the score vector under the null hypothesis can be estimated more easily by estimating only correlations between genetic covariates instead \n\n\\subsubsection{The normal model}\n\nFor $Y_i$ normally distributed, $\\Lambda=\\sigma^2I$, where $I$ is the $n\\times n$ identity matrix. The score vector can then be written\n\\[\n  \\bm U_{\\text g \\mid\\text e}=\\frac1{\\sigma^2} \\xg^T (I-H)\\y,\n\\]\nand \\eqref{var} reduces to\n\\[\n  V_{\\text g\\mid\\text e}=\\frac{1}{\\sigma^2}\\xg^T(I-H)\\xg,\n\\]\nwhere $H=\\xe(\\xe^T\\xe)^{-1}\\xe^T$ is the idempotent matrix projecting onto the column space of $\\xe$. Then $I-H$ is the idempotent matrix projecting onto the orthogonal complement of the column space of $\\xe$, and $(I-H)\\bm Y$ are the residuals when fitting the multiple linear model with only the environmental covariates present. Note that $\\sigma^2$ enters into the test statistics $T_j$~\\eqref{eq:Tk}, and needs to be replaced by an estimate; we have used the residual sum of squares of a fitted model with only environmental covariates present (the null hypothesis), divided by $n-d$.\n\n\\subsubsection{The logistic model}\n\nFor $Y_i$ Bernoulli distributed, $\\phi=1$ and the $\\sigma_i^2$ of $\\Lambda$ are estimated by $\\hat \\mu_{\\text ei}(1-\\hat \\mu_{\\text ei})$, where $\\hat \\mu_{\\text ei}$ are the fitted values under the null hypothesis with only environmental covariates. Inference about $\\bg$ is valid also if data are collected in a case--control study since the canonical (logit) link is used \\citep[pp. 170--171]{agresti2002categorical}.\n\nIn the special case of no environmental covariates, that is, $\\xe=\\bm1$, each score test statistic, $T_j$ \\eqref{Tk-noenv}, is equal to the Cochran--Armitage trend test \\citep{Armitage1955,Cochran1954} statistic,\n\\[\n \\frac{\\sum_{i=0}^2s_i(n_2x_i-n_1y_i)}\n  {\\sqrt{n_1n_2\\big(\\sum_{i=0}^2s_i^2m_i-\\frac1n(\\sum_{i=0}^2s_im_i)^2\\big)}},\n\\]\nwhere $s_i$ are the possible values of the genetic covariates, $n_1$ and $n_2$ the number of 0 and 1 phenotypes $Y_i$, respectively, $x_i$ the number of observations having phenotype 1 and genotype $s_i$ at marker $k$, \\ $y_i$ the number of observations having phenotype 0 and genotype $s_i$, and $m_i=x_i+y_i$. The Cochran--Armitage test is used in disease--genotype association testing with scores $(s_0,s_1,s_2)=(0,s,1)$ \\citep{sasieni1997genotypes,slager2001case}, for example with $s=\\frac12$ for an additive genetic model.\n\n\\section{Familywise error rate control and approximations}\\label{sec:multtest}\n\nWe now turn to the topic of how to control the familywise error rate (FWER) by intersection approximations, and then apply this to our situation.\n\n\\subsection{Multiple hypothesis familywise error rate control}\\label{sec:fwer}\n\nWe have a collection of $m$ null hypotheses, $H_k\\colon \\bg[k]=0$ (no association between phenotype and genotype at marker $k$), $1\\leq k\\leq m$, against two-sided alternatives. We will present a method for multiple testing correction that controls the FWER -- the probability of making at least one type~I error. We adopt the notation of \\cite{MoskvinaSchmidt2008}, and denote by $O_k$ the event that the null hypothesis $H_k$ is not rejected, and by $\\bar O_k$ its complement, $1\\leq k\\leq m$. Then, if all $m$ null hypotheses are true,\n\\begin{equation}\n \\text{FWER} = P(\\bar O_1 \\cup \\cdots \\cup \\bar O_m) = 1-P(O_1 \\cap \\cdots \\cap O_m). \\label{eq:FWER}\n\\end{equation}\nIn our case, $O_k$ is an event of the form $|T_k| < c$, where $T_k$ is the test statistic of \\eqref{eq:Tk}. We will consider single-step multiple testing methods, and choose the same cut-off $c$ for each $k$. We denote by $\\alpha_{\\text{loc}}=2\\Phi(-c)=P(\\bar O_k)$, the asymptotic probability of false rejection of $H_k$, where $\\Phi$ is the univariate standard normal cumulative distribution function. When the joint distribution of the test statistics is known under the complete null hypothesis, or can be estimated, FWER control at the $\\alpha$ significance level can be achieved by solving the inequality $\\text{FWER} \\leq \\alpha$ for $\\alpha_{\\text{loc}}$, based on either the union or intersection formulation of~\\eqref{eq:FWER}. When $m$ is large, this involves evaluating high dimensional integrals over the acceptance or rejection regions, which is suggested by \\cite{ConneelyBoehnke2007}.\n\nTo avoid evalulating these costly integrals, we may instead control FWER by considering bounds based on~\\eqref{eq:FWER}. For example, the Bonferroni method is based on the Boole inequality applied to the union formulation of~\\eqref{eq:FWER},\n\\[\n \\text{FWER} = P(\\bar O_1 \\cup \\cdots \\cup \\bar O_m) \\leq \\sum_{k=1}^m P(\\bar O_k)=\\sum_{k=1}^m \\alpha_{\\text{loc}}=m\\alpha_{\\text{loc}},\n\\]\nfrom which it is seen that a local significance level of $\\alpha_{\\text{loc}}=\\alpha\/m$ guarantees $\\text{FWER}\\leq\\alpha$.\n\nWhen the FWER is calculated under the complete null hypothesis, so-called weak FWER control is achieved. However, in our situation, subset pivotality is satisfied, meaning that the distribution of any subvector $(T_k)_{k\\in K}$ is identical under $\\bigcap_{k\\in K}H_k$ and under the complete null hypothesis $\\bigcap_{k=1}^mH_k$, for all subsets $K\\subseteq\\{1,2,\\ldots, m\\}$. In particular, a subvector of $\\bm U_{\\text g\\mid\\text{e}}$~\\eqref{u} and a submatrix of $V_{\\text g\\mid\\text{e}}$~\\eqref{var} corresponding to $K$ only involves genetic covariates corresponding to~$K$. Then strong FWER control is achieved, meaning that $\\text{FWER}\\leq\\alpha$ regardless of which null hypotheses are true \\citep{WestfallYoung1993,westfall2008multiple}. \n\nThe focus in this work will be on the intersection formulation of~\\eqref{eq:FWER}. Background theory will be given next and new application in \\ref{sec:FWERkapprox}.\n\n\\subsection{Intersection approximations}\\label{sec:fwerk}\n\nFollowing \\cite{glazjohnson}, we define $k$th order product-type approximations to $P(O_1\\cap\\nobreak\\cdots\\cap O_m)$ by\n\\begin{equation}\n \\gamma_k=P(O_1\\cap\\cdots\\cap O_k)\\prod_{j=k+1}^mP(O_j\\mid O_{j-k+1}\\cap\\cdots\\cap O_{j-1})\n  =\\frac{\\prod_{j=k}^{m}P(O_{j-k+1}\\cap\\cdots\\cap O_j)}{\\prod_{j=k+1}^{m}P(O_{j-k+1}\\cap\\cdots\\cap O_{j-1})},\n\\label{gammak}\n\\end{equation}\n$1\\leq k\\leq m$, where probabilities are evaluated under the complete null hypothesis. This is similar to the usual multiplicative rule for the probability of intersection of events applied to $\\gamma_m=P(O_1\\cap\\nobreak\\cdots\\cap O_m)$, but with dimension of distributions limited to $k$. The idea is that the $\\gamma_k$ should constitute increasingly better approximations of $\\gamma_m$ as $k$ increases, and that calculation of $\\gamma_k$ is less costly than calculation of $\\gamma_m$ when $k<m$, since only $k$-variate distributions are involved in $\\gamma_k$.\n\n\\sloppy\nNote that the approximations depend on the order of the components of $\\bm T=(T_1,\\ldots T_m)$. We have used the order in which the $m$ markers are positioned along the genome, assuming that the largest correlations occur between close markers.\n\n\\fussy\nIn our case, $\\gamma_1=\\prod_{j=1}^m P(\\lvert T_j\\rvert<c)=(1-\\alpha_{\\text{loc}})^m$ and $\\gamma_m=P(\\lvert T_1\\rvert<c,\\ldots,\\lvert T_m\\rvert<\\nobreak c)=1-\\text{FWER}$. Since $\\bm T$ is asymptotically multivariate normally distributed with mean~$\\bm0$ under the complete null hypothesis, $\\gamma_1\\leq\\gamma_m$ asymptotically \\citep{Sidak1967}. Choosing $\\alpha_{\\text{loc}}$ such that $\\text{FWER}=1-\\gamma_m\\leq1-\\gamma_1=1-(1-\\alpha_{\\text{loc}})^m=\\alpha$ keeps FWER at the $\\alpha$ level. It is well known that the $\\alpha_{\\text{loc}}$ found by this method, the \\v Sid\\'ak  method, is slightly larger than the $\\alpha_{\\text{loc}}$ found by the Bonferroni method, thus the \\v Sid\\'ak  method will give slightly higher power.\n\nWe have seen that in our case, $\\gamma_1\\leq\\gamma_m$, meaning that the \\v Sid\\'ak  method can safely be used. If $\\gamma_k\\leq\\gamma_m$, then $\\text{FWER}=1-\\gamma_m\\leq1-\\gamma_k=\\alpha$ can be used to control FWER by solving the last equation for $\\alpha_{\\text{loc}}$ (choosing the greatest solution if not unique -- we have, however, never observed a $\\gamma_k$ that is not monotonically decreasing in $\\alpha_{\\text{loc}}$). If $\\gamma_k\\leq\\gamma_l$, then continuity of $\\gamma_k$ and of $\\gamma_l$ as functions of $\\alpha_{\\text{loc}}$ implies that the $\\alpha_{\\text{loc}}$ making $1-\\gamma_l=\\alpha$ is no less than the $\\alpha_{\\text{loc}}$ making $1-\\gamma_k=\\alpha$, so that the power obtained by the $l$th approximation is no less than the power obtained by the $k$th approximation.\n\n\\sloppy\nThe ideal property $\\gamma_1\\leq\\gamma_2\\leq\\cdots\\leq\\gamma_k\\leq\\gamma_m$ for all $\\alpha_{\\text{loc}}$ is ensured if $\\lvert\\bm T\\rvert=(\\lvert T_1\\rvert,\\ldots,\\lvert T_1\\rvert)$ is monotonically sub-Markovian of order $k$ ($\\text{MSM}_k$) with respect to $(-\\infty,c)^k$ for all $c$, \\ $2 \\leq k \\leq m-1$, as defined by \\cite{Block1992}. Unfortunately, our $\\lvert\\bm T\\rvert$ is not $\\text{MSM}_{m-1}$. It is possible to construct a trivariate normal distribution with mean $\\bm0$ such that $\\gamma_1<\\gamma_3<\\gamma_2$ for some~$\\alpha_{\\text{loc}}$. However, the violations of $\\text{MSM}_{m-1}$ we have observed have been very small and only for restricted ranges of $\\alpha_{\\text{loc}}$, and only for carefully constructed covariance matrices. We have not observed violations for covariance matrices estimated from real data, and will therefore proceed to apply $\\gamma_2$ and $\\gamma_3$ as better approximations to $\\gamma_m$ than $\\gamma_1$ (the latter giving \\v Sid\\'ak  cutoffs). A summary of concepts of positive dependence, like $\\text{MSM}$, was given by \\citet[pp.~58--61]{Dickhaus2014}.\n\n\\fussy\n\\subsection{Controlling FWER using $k$th order approximation for score tests}\\label{sec:FWERkapprox}\n\nAs we have seen, the vector $\\bm T$ of score test statistics is under the complete null hypothesis asymptotically standard multivariate normal with covariance matrix $R$ \\eqref{eq:R}. We denote by $O_j$ the event $\\lvert T_j\\rvert<c$ of non-rejection of $H_j$, which has probability $P(O_j)=1-\\alpha_{\\text{loc}}$ under the null hypothesis, with $\\alpha_{\\text{loc}}=2\\Phi(-c)$. We will detail how to find $\\alpha_{\\text{loc}}$ given by the second order approximation, $\\gamma_2$: Denote by $r_j$ the $(j-1,j)$ entry of $R$. Then\n\\[\n P(O_{j-1}\\cap O_j)= 1-\\alpha_{\\text{loc}}-\\sqrt{\\frac{2}{\\pi}} \\int_{-c}^{c}\\!\\!e^{-x^2\/2}\\,\\Phi\\Biggl(\\frac{r_j x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Biggr)dx,\n\\]\ngiving\n\\begin{align}\n \\gamma_2&=P(O_1\\cap O_2)\\prod_{j=3}^mP(O_j\\mid O_{j-1})\n  =\\frac{\\prod_{j=2}^m P(O_{j-1} \\cap O_j)}{\\prod_{j=3}^{m} P(O_{j-1})}\\nonumber\\\\\n &= \\frac{\\prod_{j=2}^{m}\\Bigl(1-\\alpha_{\\text{loc}}-\\sqrt{\\frac{2}{\\pi}}\\int_{-c}^{c}e^{-x^2\/2}\\,\\Phi\\Bigl(\\frac{r_{j}x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Bigr)dx\\Bigr)}{(1-\\alpha_{\\text{loc}})^{m-2}}\\nonumber\\\\\n &= (1-\\alpha_{\\text{loc}})\\prod_{j=2}^{m}\\Biggl(1-\\sqrt{\\frac{2}{\\pi}}\\frac{1}{1-\\alpha_{\\text{loc}}}\\int_{-c}^{c}\\!\\!e^{-x^2\/2}\\,\\Phi\\Biggr(\\frac{r_{j}x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Biggr)dx\\Biggr).\n\\label{eq:gamma2}\n\\end{align}\nFor a desired upper bound $\\alpha$ on FWER, the equation $1-\\gamma_2=\\alpha$ is solved with respect to $\\alpha_{\\text{loc}}$, which can be done numerically using for example a bisection algorithm. Note that $\\alpha_{\\text{loc}}$ enters into $c=-\\Phi^{-1}(\\alpha_{\\text{loc}}\/2)$.\n\nWe can control FWER by higher-order approximations by solving the equation $1-\\gamma_k=\\alpha$ for $\\alpha_{\\text{loc}}$ in a similar way, which we will henceforth refer to as order $k$ FWER approximation. By~\\eqref{gammak}, $\\gamma_k$ can be written as a ratio of products of $k$-dimensional and products of $k-1$-dimensional multivariate normal integrals. Good numerical methods for calculating multivariate normal integrals exist for small dimensions \\citep{genz2009computation}. We will illustrate using $k=2$ and $k=3$ for real data in Section \\ref{sec:invest}.\n\nThe procedure to find $\\alpha_{\\text{loc}}$ does not depend on the exact form of the test statistic, only that the vector $(T_1,\\ldots,T_m)$ of test statistics is asymptotically standard multivariate normal under the complete null hypothesis and $\\lvert T_j\\rvert\\geq c$ leads to rejection. In particular, \\eqref{eq:gamma2} is identical to what was found by \\cite{MoskvinaSchmidt2008} for an allelic test and correlations given by linkage disequilibria.\n\nIn practice, instead of calculating $\\alpha_{\\text{loc}}$, it may be preferable to calculate FWER-adjusted $p$-values: Replace $\\alpha_{\\text{loc}}$ with $p$, the unadjusted $p$-value for an individual test, in the calculation of $\\gamma_k$. Then $1-\\gamma_k$ is an FWER-adjusted $p$-value for the test, in the sense that if $1-\\gamma_k\\leq\\alpha$ (rejection based on adjusted $p$-value), then $p\\leq\\alpha_{\\text{loc}}$ (rejection based on local significance level).\n\n\\subsection{FWER control with independent blocks}\\label{sec:blocks}\n\nGenetic markers are distributed along the chromosomes and a common assumption is independence of genetic markers from different chromosomes.\n\nAs we have seen in Section \\ref{sec:corr}, if the genetic markers are independent and no environmental covariates that are correlated with the genetic markers are included, the score test statistics for these markers would also be independent. Within a chromosome, genetic markers can belong to different haplotype blocks, being highly correlated within a block and independent or nearly independent between the blocks \\citep{biologi}.\n\nAssume that the $m$ markers to be tested, and thus $\\{O_1,\\ldots,O_m\\}$, can be partitioned into $b$ independent blocks, $\\{O_1,\\ldots,O_{m_1}\\}$, \\ $\\{O_{m_1+1},\\ldots,O_{m_2}\\}$, \\ldots, $\\{O_{m_{b-1}+1},\\ldots,O_m\\}$,  so that $O_{j_1}$ and $O_{j_2}$ are independent if they belong to different blocks. Let $\\gamma_k^{(l)}$ be the $k$th order approximation given by~\\eqref{gammak} for the intersection of the events belonging to the $l$th block, \\ $1\\leq l\\leq b$, and let $\\gamma_k$ be the overall $k$th order approximation. Then it is easy to verify that $\\gamma_k=\\prod_{l=1}^b\\gamma_k^{(l)}$.\n\n\\subsection{The effective number of independent tests}\\label{sec:Meff}\n\nThe concept of an effective number of independent tests, $M_{\\text{eff}}$, in multiple\ntesting problems has been described and discussed by many authors, including \\cite{nyholt}, \\cite{gao1}, \\cite{MoskvinaSchmidt2008}, \\cite{LiJi2005}, \\cite{Galwey2009} and \\cite{ChenLiu2011}. All except \\cite{MoskvinaSchmidt2008} first estimate $M_{\\text{eff}}$, and then use $M_{\\text{eff}}$ in place of $m$ in the \\v Sid\\'ak  formula to calculate $\\alpha_{\\text{loc}}=1-(1-\\alpha)^{1\/M_{\\text{eff}}}$. An alternative formulation using the Bonferroni formula also exists.\n\nNone of these methods use the concept of FWER in the derivation of $M_{\\text{eff}}$, and there is no mathematical justification that FWER is controlled. All methods start with the linkage disequilibrium or composite linkage disequilibrium matrix, and there is no mention of the dependence of the $M_{\\text{eff}}$ estimate on the test statistics used for the hypothesis tests.\n\nThe method of \\cite{MoskvinaSchmidt2008} is based on an allelic test and controls the FWER using second order intersection approximations. As for our method, the main output of their method is an estimate of $\\alpha_{\\text{loc}}$. The above \\v Sid\\'ak  formula can then be used to define $M_{\\text{eff}}=\\ln(1-\\alpha)\/\\ln(1-\\alpha_{\\text{loc}})$. Note that $M_{\\text{eff}}$ depends on both $\\alpha_{\\text{loc}}$ and the FWER threshold~$\\alpha$. We will not consider $M_{\\text{eff}}$ further in this article. \n\n\\subsection{The maxT permutation method}\\label{maxT}\n\nWe will compare the local significance level $\\alpha_{\\text{loc}}$, as calculated by the FWER approximation method presented in section~\\ref{sec:FWERkapprox}, with the \\cite{WestfallYoung1993} maxT permutation method, and give a brief review of the latter.\n\nThe FWER is the probability that at least one of the $m$ null hypotheses is falsely rejected, which can be formulated as $P(\\max_j\\lvert T_j\\rvert\\geq c)=1-\\gamma_m$ under the complete null hypothesis. In the maxT method, the critical value $c$ is found empirically by permutation of the response variable in order to generate a sample from the distribution of the $\\max_j\\lvert T_j\\rvert$ statistic. If the FWER is to be controlled at the $\\alpha$ level and $b$ permutations are made, $c$ is estimated by the $(1-\\alpha)b$th order statistic of the $\\max_j\\lvert T_j\\rvert$ (the $(1-\\alpha)b$th smallest value), which is an estimate of the $1-\\alpha$ quantile of $\\max_j\\lvert T_j\\rvert$. The probability that the $k$th order statistic of a random sample of size $b$ is greater than the $1-\\alpha$ quantile is equal to the binomial cumulative distribution function with parameters $b$ and $1-\\alpha$ evaluated at $k-1$, which can be used to construct a confidence interval for $c$ (\\citeauthor{thompson1936confidence}, \\citeyear{thompson1936confidence}, see e.g. \\citeauthor{conover1980practical}, \\citeyear{conover1980practical}, p. 114). A confidence interval for $\\alpha_{\\text{loc}}$ is obtained by transforming the bounds via $\\alpha_{\\text{loc}}=2\\Phi(-c)$.\n\nThe success of the permutation method relies on the exchangeablity of the data, which in general does not hold for regression problems \\citep{Commenges2003}.  In our GLM the responses $\\bm Y$ are in general not exchangeable since their expected values are not equal when environmental covariates (which may not be independent of the genetic covariates) are present. Without environmental covariates (only intercept) the responses are exchangeable and permutation of $\\bm{Y}$ gives FWER control. With discrete environmental covariates permutation can be done in a stratified manner \\citep{Solari2014}. When the exchangeability assumption is not satisfied, there is no standard solution to how permutation testing can be performed. Asymptotic or second moment exchangeability may be obtained by different transformations of the data, but comparison with these methods is beyond the scope of this paper. \n\n\\section{Power and efficiency of the FWER approximation method}\\label{sec:invest}\n\nWe will compare the local significance level $\\alpha_{\\text{loc}}$, as calculated by the FWER approximation method presented in the previous section, with $\\alpha_{\\text{loc}}$ of the Bonferroni method and of the \\cite{WestfallYoung1993} maxT permutation method. We proceed to compare the $\\alpha_{\\text{loc}}$ calculated by FWER approximation in two cases were the ``true'' $\\alpha_{\\text{loc}}$ based on the entire joint distribution can be calculated; one artificial and one based on data.\n\n\\subsection{Illustration of methods: TOP and $\\text{VO}_2$-max data}\\label{sec:topvo2}\n\nOur two data sets (referred to as TOP and $\\text{VO}_2$-max) are of limited sample size, and our aim is to use the data to investigate the correlation structure of GWA-data and the effect this has on the estimation of the local significance level. We assume that our findings will hold in data sets with larger sample sizes. An increase in sample size will give more precise estimates of the score test statistics correlations, but the estimation of the local significance level is mainly dependent on score test statistics correlations under study (not the sample size).\n\nThe TOP data set is a case--control GWA data set, in which case is schizophrenia or bipolar disorder. The data set was collected with the aim to detect single-nucleotide polymorphisms (SNPs) associated with the schizophrenia or bipolar disorder \\citep{TOP1,TOP2}. The preprocessed TOP GWA data contain genetic information on 672972 SNPs (Affymetrix Genome-Wide Human SNP Array 6.0) for 1148 cases and 420 controls. Our dataset included individuals sampled until March 2013, and therefore the sample size is larger than in the cited papers. Preprosessing of the data was done as described in \\cite{TOP1} and \\cite{TOP2}.\n\nGenotype--phenotype association was assessed by fitting a logistic regression without any environmental covariates, so that score test correlations equal genotype correlations (Section~\\ref{noenvcov}).\n \nThe $\\text{VO}_2$-max data set comes from a cross-sectional GWA study \\citep{VO2max1, VO2max2}, in which the aim was to find SNPs associated with maximum oxygen uptake. The preprocessed $\\text{VO}_2$-max GWA data consist of 123497 SNPs \\citep[Illumina Cardio-MetaboChip,][]{VO2chip} for 2802 individuals. The $\\text{VO}_2$-max data were analysed using a normal linear regression model, including age, sex and physical activity score as covariates. For both datasets, some genotype data were missing. In the TOP data, mean imputation was done for $0.04\\%$ of the genotypes, and in the $\\text{VO}_2$-max data for $0.7\\%$ of the genotypes. \n\nFor the $\\text{VO}_2$ data, the local significance level controlling the FWER at level 0.05 was lowest for the Bonferroni method, slightly higher for the order 1 approximation (the \\v Sid\\'ak method), and further increasing through the order 2 and~3 FWER approximations (Table~\\ref{tab:resultater}).\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccccc}\\hline\n & \\multicolumn2c{TOP} && \\multicolumn2c{$\\text{VO}_2$-max}\\\\ \\cline{2-3}\\cline{5-6}\nMethod & $10^8\\alpha_{\\text{loc}}\\vphantom{\\big)}$ & Ratio & & $10^7\\alpha_{\\text{loc}}$ & Ratio \\\\\n\\hline\nBonferroni  & 7.43 & 1.00 && 4.05 & 1.00\\\\\nOrder 1 (\\v Sid\\'ak)   & 7.62 & 1.03 && 4.15 & 1.02\\\\\nOrder 2 & 8.62 & 1.16 && 4.70 & 1.16 \\\\\nOrder 3 & 9.07 & 1.22 && 5.02 & 1.24 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ calculated by the Bonferroni method and by order 1--3 FWER approximations for the TOP and $\\text{VO}_2$-max data, controlling the FWER at level $0.05$, \nand ratio of $\\alpha_{\\text{loc}}$ to Bonferroni $\\alpha_{\\text{loc}}$.}\n\\label{tab:resultater}\n\\end{center}\n\\end{table}\n\nFor the TOP data, since no environmental covariates are included, permutation of the binary response vector is feasible\n(the exchangeability assumption is satisfied), and the maxT method can be used to estimate the local significance level controlling FWER at level $0.05$. Permutation of the responses, followed by calculation of the maximal score test statistics over the whole genome, is a time consuming task, and we will only present results on two of the smallest chromosomes (chromosome 21 and 22):\n\nThe local significance level controlling the FWER at level 0.05 was, as for the $\\text{VO}_2$ data, lowest for the Bonferroni method and increasing through order 1--3 FWER approximations (Table \\ref{tab:TOPres}). The highest level was obtained for the maxT method, and also the lower bound of the 95\\% confidence interval for $\\alpha_{\\text{loc}}$ of maxT was greater than the order 3 FWER approximation.\n\nOn a $4\\times6$-core Xeon 2.67 GHz computer (Intel CPU) running Linux (Ubuntu 14.0) using one thread, the analyses on chromosome 22 took 85 hours for maxT, 20 minutes for order 3 FWER and 10 seconds for order 2 FWER approximation.\n\nSmoothed frequency distributions of the estimated correlations between neighbouring SNPs along chromosomes are very similar across chromosomes (Figure \\ref{fig:densTOPandVO2}), and therefore, we would expect that the trends for chromosome 21 and 22 can be extended to the other chromosomes and to the whole genome.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{.\/neighbourcorr-top-4}\\hfill\n\\includegraphics[width=0.49\\textwidth]{.\/neighbourcorr-vo2-4}\n\\caption{Smoothed frequency distributions of absolute value of the estimated genotype correlations between neighbouring SNPs on each chromosome, one line per chromosome. TOP data (left) and $\\text{VO}_2$-max data (right). The plots are logspline density estimates \\citep{stone1997polynomial}, implemented by the R logspline package \\citep{logsplineR}.}\n\\label{fig:densTOPandVO2}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccccc}\\hline\n & \\multicolumn5c{Chromosome}\\\\ \\cline{2-6}\n & \\multicolumn2c{21} && \\multicolumn2c{22}\\\\ \\cline{2-3}\\cline{5-6}\nMethod & $10^6\\alpha_{\\text{loc}}\\vphantom{\\big)}$ & Ratio && $10^6\\alpha_{\\text{loc}}$ & Ratio\\\\\n\\hline\nBonferroni & 5.10 & 1.00 && 5.57 & 1.00\\\\\nOrder 1 (\\v Sid\\'ak) & 5.23 & 1.03 && 5.72 & 1.03\\\\\nOrder 2 & 6.09 &1.19 && 6.46 & 1.16\\\\\nOrder 3 & 6.57 & 1.29 && 6.93 &1.24\\\\\nmaxT, lower & 7.34 & 1.44 && 7.68 & 1.38\\\\\nmaxT & 7.44 &1.46 && 7.79 &1.40\\\\\nmaxT, upper & 7.55 & 1.48 && 7.91 & 1.42\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ for the TOP data calculated by the Bonferroni method and by order 1--3 FWER approximations, and estimated by the maxT method, controlling FWER at level $0.05$, and ratio of $\\alpha_{\\text{loc}}$ to Bonferroni $\\alpha_{\\text{loc}}$. Chromosome 21 contained $9802$ SNPs and chromosome~22 contained $8970$ SNPs. The number of permutations for the maxT method was 500000. The lower and upper values for maxT are bounds of a 95\\% confidence interval for $\\alpha_{\\text{loc}}$ (see Section~\\ref{maxT}).}\n\\label{tab:TOPres}\n\\end{table}\n\n\\subsection{Correlation structure and local significance level}\\label{sec:alphaksim}\n\nConsider 100 markers and a multivariate normal test statistic $\\bm T$ having an AR1 correlation structure, that is, all entries on the main diagonal of the $100\\times100$ correlation matrix are equal to~1, on the sub- and superdiagonal $\\rho$, on the next diagonals $\\rho^2$, and so on. We investigated the effect of positive $\\rho$ on the local significance level $\\alpha_{\\text{loc}}$ found by order 1--4 FWER approximations to control FWER at the 0.05 level. Also, the ``true'' $\\alpha_{\\text{loc}}$ was calculated without approximation (that is, based on $\\gamma_{100}$; see Section~\\ref{sec:fwerk}), using the \\texttt{pmvnorm} function of the R \\citep{R} package mvtnorm \\citep{mvtnormR} using the Genz--Bretz algorithm \\citep{Genz1992,Genz1993,GenzBretz2002}. The \\texttt{pmvnorm} function can calculate multivariate normal probabilities with some accuracy for dimensions up to 1000.\n\nThe inverse of an AR1 correlation matrix contains only negative off-diagonal entries, which ensures a property called  $\\text{MTP}_2$ \\citep{karlin1981total} for the density of $\\lvert\\bm T\\rvert$, which implies that the product-type approximations $\\gamma_k$ of Section~\\ref{sec:fwerk} are non-decreasing in $k$ \\citep{glazjohnson}, making the $\\alpha_{\\text{loc}}$ of the order $k$ FWER approximations non-decreasing in $k$.\n\nThe effect of $\\rho$ on $\\alpha_{\\text{loc}}$ was small for $\\rho<0.4$ (Figure \\ref{fig:ar1}), so for the 100 markers considered, there would be no gain in using FWER approximation or even the true joint distribution of $\\bm T$ instead of \\v Sid\\'ak this case. For larger $\\rho$, order 2 FWER approximation provides an improvement compared to \\v Sid\\'ak. The increase in $\\alpha_{\\text{loc}}$ from \\v Sid\\'ak to order 2 FWER approximation was greater than the difference between higher orders.\n\nTo assess the order $k$ FWER approximation method for a more realistic correlation structure, we considered the empirical correlation matrix for the first 1000 markers on chromosome 22 of the TOP data. The order 1--4 approximations to control FWER at the 0.05 level gave an $\\alpha_{\\text{loc}}$ of 5.1 (\\v Sid\\'ak), 5.8, 6.2 and $6.4\\cdot10^{-5}$, respectively, whereas the $\\alpha_{\\text{loc}}$ calculated without approximation using the Genz--Bretz algorithm was $7.3\\cdot10^{-5}$.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.49\\textwidth]{.\/ar1.pdf}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ for order 1--4 FWER approximations and $\\alpha_{\\text{loc}}$ based on true joint distribution of test statistic as a function of the parameter $\\rho$ of an AR1 correlation matrix for 100 markers. The horizontal line corresponds to \\v Sid\\'ak correction (order 1 FWER approximation), then $\\alpha_{\\text{loc}}$ is increasing with the order of the approximation (order 2--4; the three curves in the middle). The uppermost curve shows $\\alpha_{\\text{loc}}$ based on the true joint distribution.}\n\\label{fig:ar1}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discuss}\n\nWe have presented the order $k$ FWER approximation method for estimating the local significance level $\\alpha_{\\text{loc}}$ used to control FWER in a GWA study.  Our method takes the estimated correlation structure between the test statistics into account, and is applicable when environmental covariates are present. The relation between the phenotype response and the genetic and environmental covariates can be modelled by any generalized linear model (using the canonical link); in particular, both models with discrete and models with continuous phenotypes are allowed. We have applied the method to common genetic variants, but it can also be used for rare variants. However, since rare variants are less correlated than common variants, we expect the increase in $\\alpha_{\\text{loc}}$ from the \\v Sid\\'ak  method to be less than when analyzing common variants. \n\nThe order $k$ FWER approximation is based on conditioning on the previous $k-1$ neighbouring markers along the chromosome. A sufficient condition to have non-decreasing local significance levels when the order of the FWER approximation increases from 1 to $k$, and that the order $k$ order approximation gives valid FWER control, is that the test statistic has the $\\text{MSM}_k$ property. Even $\\text{MSM}_2$ and $\\text{MSM}_3$ are difficult to verify for our test statistic with GWA data, but it is reasonable to assume that they are satisfied (Section \\ref{sec:fwerk}).\n\nPopulation substructure can be associated with both the genotype and phenotype and is therefore a possible confounding factor in GWA studies. Population substructure can be adjusted for in the analysis using principal components of the covariance matrix of the individuals \\citep{Price2006} as covariates. In both the TOP data and the VO$_2$-max data, related individuals were removed in the preprocessing of the data, and no adjustment for population structure was done in our analysis. \n\nThe AR1 correlation structure (Section~\\ref{sec:alphaksim}) might not be a realistic model for genotype correlations, but the calculations nevertheless show potential for a significant improvement over the \\v Sid\\'ak method by applying the fast order 2 FWER approximation. Also, there is potential to get quite close to the local significance level given by the full joint distribution of the test statistic vector by using order 3 or 4 approximation. Calculations using the more realistic empirical correlation matrix of part of chromosome 22 of the TOP study confirm this impression.\n\nThe maxT method (Section~\\ref{maxT}) of \\cite{WestfallYoung1993} may give higher power than FWER approximation (Table~\\ref{tab:TOPres}). However, there is no general way of including environmental covariates using that method (Section~\\ref{maxT}). Also, computing time is much larger than for lower-order FWER approximation (Section~\\ref{sec:topvo2}), and the $\\alpha_{\\text{loc}}$ estimate would likely differ if a new set of permutations were made (see confidence limits of Table~\\ref{tab:TOPres}).\n\nAnother alternative is parametric bootstrap methods \\citep{SeamanMyhsok2005}, which  could be used to estimate the local significance level when the exchangeability assumption is not satisfied. It would be an efficient method, but to our knowledge it has not been proven that parametric bootstrap will control the overall error rate, since nuisance parameters need to be estimated.\n\n\\cite{ConneelyBoehnke2007} introduced a method for multiple testing correction for GLMs for multiple responses (traits) based on the estimated correlation matrix of the score vector. The focus of the method is to calculate FWER-adjusted $p$-values based on the multivariate integral arising from \\eqref{eq:FWER}. Currently, this integral can be computed numerically with some accuracy for dimensions smaller than or equal to 1000 using the \\texttt{pmvnorm} function of the R package mvtnorm (see Section~\\ref{sec:alphaksim} for details and references). Thus, the method of \\citeauthor{ConneelyBoehnke2007} is not applicable for larger problems, e.g. more than $m=1000$ hypothesis tests. \n\nFor our order $k$ FWER approximation method we have used standard R functions to compute the second order approximation given by~\\eqref{eq:gamma2}. For orders 3, 4 and 5 we have used the above-mentioned function \\texttt{pmvnorm} specifying the Miwa algorithm \\citep{Miwa2003} instead of the default Genz--Bretz algorithm. The Miwa algorithm can be used for small dimensions, and is deterministic, whereas the Genz--Bretz algorithm includes simulations that lead to inaccuracies, which accumulate to an intolerable level when used for the large number of factors in~\\eqref{gammak}. The research into better and faster integration of multivariate normal densities is ongoing, and \\cite{Botev2016} provides an interesting new approach, applicable for dimensions smaller than or equal to 100. This will enable our order $k$ FWER approximation method to be applied with larger values of~$k$ than what has been presented here.\n\n\\section{Conclusions}\\label{sec:conclude}\nWe have presented a new method for controlling the FWER for GWA data. The order~$k$ FWER approximation method can be used for generalized linear models and include adjustment for environmental covariates, possibly confounding, like population substructure or sex. We have applied the FWER approximation method to GWA data, and shown that our method is a powerful alternative to the Bonferroni and \\v Sid\\'ak  methods, especially in situations were permutation methods cannot be used (exchangeability assumption not satisfied).\n\nThe method provides a local significance level, $\\alpha_{\\text{loc}}$, for the individual tests, meaning that the null hypothesis of no association between phenotype and genetic marker should be rejected if the (unadjusted) $p$-value of a test is less than $\\alpha_{\\text{loc}}$. We found a substantial increase in $\\alpha_{\\text{loc}}$ already at the order~2 approximation, compared to the $\\alpha_{\\text{loc}}$ produced by the well-known Bonferroni and the \\v Sid\\'ak methods -- methods that does not take correlation structure between the test statistics of the markers into account (\\v Sid\\'ak assumes independence, but that could be considered worst-case for GWA data).\n\n\\section*{Software}\nThe statistical analysis were performed using R \\citep{R}, and the preprocessing of the genetic data were done using the software PLINK \\citep{plink}.  \n\n\\section*{Acknowledgements}\nThe authors would like to thank Dr. Jelle J. Goeman (Leiden University Medical Centre, Leiden, The Netherlands) for valuable comments. Part of the work was done while the last author was on sabbatical at Centre for the Genetic Origins of Health and Disease, University of Western Australia, Australia. The PhD position of the first author is founded by the Liaison Committee between the Central Norway Regional Health Authority (RHA) and the Norwegian University of Science and Technology (NTNU).\n\n\\emph{Conflict of Interest:} None declared.\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUltrarelativistic heavy-ion experiments have shown very rich physics \nwhich cannot be interpreted by mere extrapolation from elementary \nnucleon-nucleon collisions to nucleus-nucleus collisions. This is evidenced from\nthe suppression of high transverse momentum region of hadron spectra up to \na factor of 5 relative to nucleon-nucleon collisions,\nwhich is an indication for strong absorption of high-energy partons traversing \nthe medium~\\cite{Wang}. Also the inclusive production of charm quark bound \nstates are suppressed by a factor of 3-5 at both Super \nProton Synchrotron (SPS) and Relativistic Heavy Ion Collider (RHIC) experiments \nwhich hints their dissolution in the medium~\\cite{Rapp,Kluberg}.\nThese accumulated evidences indicate that a new form of matter\nhas been produced in the heavy-ion collision experiments, called quark-gluon \nplasma (QGP). Among different experimental \nobservations which may be served as the signals for the QGP formation, \nquarkonium suppression has been proposed long time ago as a clear probe of \nthe QGP formation in the collider experiments~\\cite{McLer,adam05}. \nFirst, in a pioneering work by Matsui and Satz~\\cite{matsui} and \nlater in a follow up quantitative calculation~\\cite{Karsch}, it\nwas shown that the suppression of $J\/\\psi$ yields could be explained\nby the (color) screening of the potential between a heavy quark and anti-quark \nby the surrounding deconfined light quarks and gluons. \nTheoretically, one can study the quarkonium states\nby using the effective field theories. Since the mass of heavy quark, $m_Q$\nis much larger than the intrinsic scale in the theory of \nQuantum Chromodynamics (QCD), $\\Lambda_{\\rm{QCD}}$, the \nheavy quark and anti-quark are expected to move slowly with \na relative velocity $v \\ll 1$ and results in the non-relativistic version of  \nQCD (NRQCD) \\cite{nrqcd1,nrqcd2}. However, NRQCD\ndoes not fully exploit the smallness of $v$ which further gives rise \nanother effective theory, known as potential non-relativistic QCD (pNRQCD)~\\cite{pNRQCD1,pNRQCD2}\nby integrating out the momentum scale.\n\nThe heavy quark pairs formed\nin relativistic nuclear collisions develop into the physical resonances \nand traverse the plasma and then hot hadronic matter\nbefore decaying into dilepton pairs. Even before the resonance is formed\nit may be absorbed by the nucleons streaming past it~\\cite{GH} and by the\ntime the resonance is formed, the screening of the color force\nin the plasma may inhibit the formation of bound states.\nThe resonance(s) could also be dissociated either\nby hard gluons~\\cite{xu,gdiss1,gdiss2,gdiss3,gdiss4} or by comoving\nhot hadrons~\\cite{vogt}. In order to disentangle these sequential\neffects~\\cite{dks}, we must\nknow how the properties of quarkonium states change in medium.\nThe basic tools of phenomenological approach to study the properties of \nquarkonium states are potential models where the possible relativistic effects \nfor excited states of charmonium may also be incorporated there.\nAt zero temperature, the potential model\nhas made great success. At finite temperature, \nthe essence of the potential model in the context of deconfinement is to use a\nfinite temperature extension of the potential. Quantitative understanding\nof the bound state properties needs the exact potential at finite temperature \nwhich, in principle, should be derived directly from QCD, like the Cornell\npotential at zero temperature has been derived from pNRQCD from the \nzeroth-order matching coefficient. Such derivations\nat finite temperature for weakly-coupled plasma have been recently come up in the \nliterature~\\cite{Brambilla05,Brambilla08} but they\nare, however, plagued by the existence of temperature-driven \nhard as well as soft scales, $T$, $gT$, $g^2 T$, respectively.\nDue to these difficulties in finite temperature extension in effective\nfield theories, the lattice-based potentials become popular. \nHowever, neither the free energy nor the internal energy \ncan be directly used as the potential. \nIn fact, what kind of screened potentials should be used \nin the Schr\\\"odinger equation which describes well the bound states at finite \ntemperature are still an open question. \nHowever, recently more involved calculations of quarkonium spectral \nfunctions and meson current correlators obtained from potential \nmodels have been performed and compared to \nfirst-principle QCD calculations performed numerically on \nlattices~\\cite{Asak04,Dat04,Hatsuda,Jakovac,Aarts11}.\nIn addition to the uncertainties of the correct form of the\nfinite-temperature potential, there are also arbitrariness of \nthe criteria of dissociation. Besides the binding energy \nfor a particular potential as an criterion of dissociation, the decay width \nis another important quantity \nto determine the dissociation of the bound states. \nThe calculation based on a real-valued potential model does not include the \ntrue width of a state. By performing \nan analytical continuation of the Euclidean Wilson loop to Minkowski space, \nthe potential has an imaginary part due to Landau damping which clearly \nbroadens the peak~\\cite{Brambilla08,Laine1} and facilitates the early\ndissociation of the bound states.\n\nIn the RHIC or LHC era (small $\\mu_{B}$), recent lattice studies have \nconfirmed that the transition from\nnuclear matter to QGP is not a phase transition, rather a\ncrossover\\cite{phaseT}.\nThe large distance property of the heavy quark\ninteraction is important for our understanding of the bulk properties of\nthe QCD plasma phase, {\\it e.g.} the screening property of the quark gluon\nplasma \\cite{Kar62+70}, the equation of state \\cite{Dum05,Dum2} etc. \nIn these studies, deviations from perturbative \ncalculations and the ideal gas behavior are found\nbeyond the deconfinement temperature. \nIt is then reasonable to assume that the string-tension\ndoes not vanish abruptly at the deconfinement point~\\cite{string1,string2,string3}, so\none should study its effects on heavy quark potential even above $T_c$.\nThis issue, usually overlooked in the\nliterature where only a screened Coulomb potential was\nassumed above $T_c$ and the linear\/string term was assumed zero,\nwas certainly worth investigation.\nRecently a heavy quark potential at finite\ntemperature was derived by correcting the full Cornell potential,\nnot its Coulomb part alone,\nwith a dielectric function encoding the effects of the deconfined \nmedium~\\cite{prc_vineet}. This was found to have an additional\nlong range Coulomb term, in addition to the conventional Yukawa term. \nIn the short distance limit, the potential is reduced to vacuum potential, \n{\\it i.e.}, $Q \\bar Q$ pair does not see the medium, giving rise the duality\nbetween $V(r,T=0)$ and $V(0,T)$. On the\nother hand, in the large distance limit (where the screening\noccurs), potential is reduced to a long-range Coulomb potential with\na dynamically screened-color charge. Thereafter the binding energies and \ndissociation temperatures\nof the ground and the lowest-lying states of charmonium and bottomonium\nspectra have been determined~\\cite{prc_vineet,chic_vineet} which \nmatches with the finding of recent works based on\npotential models~\\cite{cabrera,mocsyprl,digal} with the Debye mass\nextracted from the lattice free energy. However, when the Debye mass \nin leading-order was used, the results~\\cite{prc_vineet}  \nmatch with the lattice correlator studies or \nwith the stronger (lattice) binding potential, i.e. internal energy~\\cite{wong1,wong2}.\nIn a way, their findings address the reason of arbitrariness of the results \non dissociation temperatures~\\cite{Satz} and ensues a basic question about the nature of dissociation\nof quarkonium in a hot QCD medium.\n\n\nHowever, all the works described above was limited to an isotropic \nmedium but the partonic system generated in an ultra-relativistic \nheavy-ion collisions cannot be homogeneous and isotropic\nbecause at the very early time of collision, asymptotic weak-coupling enhances\nthe longitudinal expansion substantially than the radial expansion so\nthe system becomes colder in the longitudinal direction than in the \ntransverse direction. As a result, an anisotropy in the momentum space sets in and \ncauses the parton system produced\nunstable with respect to the chromomagnetic plasma modes~\\cite{Romatschke} \nwhich facilitate to isotropize the system~\\cite{Arnold05,Mrowczy93}.\nIn our work, we restricted ourselves to a {\\em weakly anisotropic medium}\nbecause by the time ($t_F$ =$\\gamma \\tau_F$, $\\tau_F$ is the quarkonium \nformation time in its rest frame) quarkonia are formed in the plasma, the \nplasma becomes almost equilibrated. Motivated with this preamble on\nthe anisotropy generated in the very stage of collision, we wish to\ninvestigate the effect of weak anisotropy on the heavy-quark potential\nand subsequently on the dissociation of quarkonia states in an anisotropic\nmedium.\n\n\nOur work is organized as follows. In Section~\\ref{poten}, we will discuss the \nmedium modifications to a heavy quark potential both in isotropic and\nanisotropic medium. In Sec.\\ref{iso}, we start with the in-medium modification\nto the heavy quark potential in an isotropic medium and then \nextend it to a medium which exhibits a local anisotropy in momentum \nspace in Sec.\\ref{aniso}. \nTo do that, we first obtain the self-energy tensor for an \nanisotropic medium to obtain the hard-loop resumed \ngluon propagator. Thereafter the dielectric permittivity be \nobtained\nin terms of retarded gluon propagator and its Fourier transform at vanishing frequency \ngives the desired non-relativistic potential at finite temperature. The\npotential thus obtained depends not only on the relative separation\nof $Q \\bar Q$ pair but also on their relative orientation\nwith respect to the direction of anisotropy and \nis found always deeper than in an isotropic medium.\nIt is found that in the weak-anisotropy limit, the correction\narising due to anisotropy to the isotropic part of the potential is small\nand thus has been treated as a perturbation.\nSo, using the first-order perturbation theory, \nwe estimate the shift in energy eigen values due to the small \nanisotropic correction \nto the energy eigen values from the spherically-symmetric\npart in isotropic medium and determine their dissociation \ntemperatures in Sec.3. Finally, we conclude \nin Section~\\ref{conclu}.\n\n\\section{Heavy-quark effective potential}\\label{poten}\n\\subsection{For isotropic medium {\\texorpdfstring{$(\\xi = 0)$}{xi=0}}}\\label{iso}\n\nPotential models are based on the assumption that the\ninteraction between a heavy quark and its anti-quark\ncan be described by a potential.\nAt $T=0$, the hierarchy of well separated energy-scales: \n$m_Q\\gg m_Qv\\gg m_Qv^2$,\nallows one to systematically integrate out the different\nscales and obtain the non-relativistic potential\nQCD (pNRQCD) where the Cornell potential indeed shows up as the\nzeroth-order matching coefficient~\\cite{pNRQCD1,pNRQCD2}. Inspired \nby its success at\nzero temperature, the potential model has been applied at finite\ntemperature, with the main assumption that medium effects can be accounted \nby a temperature-dependent potential.\n\n\nRecent lattice results~\\cite{phaseT} indicate the phase transition in full QCD\nappears to be a crossover rather than a `true' phase transition with the \nrelated singularities in thermodynamic observables.\nIn light of the above findings, one cannot simply\nignore the effects of string tension between the quark-anti-quark pairs\nbeyond $T_c$. This is indeed a very important effect which needs\nto be incorporated while setting up the criterion for the dissociation.\nRecently this issue\nhas successfully been addressed for the dissociation of quarkonium in\nQGP by one of us~\\cite{prc_vineet,chic_vineet}\nand we closely follow their work in brief.\nLet us now start with a heavy quark potential (Cornell potential) at T=0 \n\\begin{equation}\n\\label{eqc}\n V(r)=-\\frac{\\alpha}{r}+\\sigma r \\quad ,\n\\end{equation}\nwhere $\\alpha$ and $\\sigma$ are the phenomenological parameters. The former\naccounts for the effective coupling between a heavy quark and its anti-quark\nand the latter gives the string coupling.\nThe medium modification enters through the Fourier transform of\nheavy quark potential as \n\\begin{equation}\n\\label{eq1}\n\\tilde{V}(k)=\\frac{V(k)}{\\epsilon(k)} \\quad ,\n\\end{equation}\nwhere $V(k)$ is the Fourier transform (FT) of the Cornell potential\nwhich requires a regularization. We regulate both terms in the\npotential by multiplying with an exponential damping factor and\nis switched off after the FT is evaluated. \nThis can be done by assuming $r$- as distribution\n($r \\rightarrow$ $r \\exp(-\\gamma r))$. The FT of\nthe linear part $\\sigma r\\exp{(-\\gamma r)}$ is\n\\begin{eqnarray}\n\\label{eq-6-3}\n=-\\frac{i}{k\\sqrt{2\\pi}}\\left( \\frac{2}{(\\gamma-i k)^3}-\\frac{2}{(\\gamma+ik)^3}\n\\right).\n\\end{eqnarray}\nAfter putting $\\gamma=0$,  we obtain the FT of the linear term $\\sigma r$ \nas,\n\\begin{equation}\n\\label{eq-6-4}\n\\tilde{(\\sigma r)}=-\\frac{4\\sigma}{k^4\\sqrt{2\\pi}}\n\\end{equation}\nand for the full Cornell potential, the FT is \n\\begin{equation}\n\\label{vk}\n{V}(k)=-\\sqrt{(2\/\\pi)} \\frac{\\alpha}{k^2}-\\frac{4\\sigma}{\\sqrt{2 \\pi} k^4}.\n\\end{equation}\nThe dielectric permittivity, $\\epsilon(k)$ is given in terms\nof the static limit of the longitudinal part of the gluon\nself-energy~\\cite{schneider}. In isotropic case, it can be \ndecomposed into longitudinal ($\\Pi_L$) and transverse ($\\Pi_T$) \ncomponents which, in the static limit, are associated with \nthe screening of electric and magnetic fields, respectively. However, in the\nstatic limit, the transverse part\n$\\Pi_T (0,k\\rightarrow 0,T)$ vanishes, {\\em i.e.}, static magnetic \nfields are not screened.\nIn perturbation theory, the quantity that enters in the Fourier transform \nof the potential at finite temperature is the static limit of\nthe longitudinal gauge boson self-energy which\nwas calculated long ago~\\cite{Shur78} at one loop level \n\\begin{eqnarray}\n\\lim_{k\\rightarrow 0} \\Pi_L (0,k,T)=g^{2} T^{2}\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)\n\\equiv m_{{}_D}^2 \\quad ,\n\\label{debye}\n\\end{eqnarray}\nwhere $ m_{{}_D} $ is defined as the screening mass. Since the static \nlimit of the \nself-energy is momentum independent, the pole of the inverse of dielectric\npermittivity is simply the gauge invariant Debye \nmass $m_{{}_D} $, so it leads to an exponential damping of the\npotential $V(r)\\sim \\exp(-m_{{}_D}r)\/r$. In particular, this form of $ \\Pi_L $\nhas the consequence that gluons screen the strong interaction, in contrast \nto the zero temperature case, over long-distance scale. If one\nassumes non-perturbative effects such as the string\ntension which survives even above the deconfinement point~\\cite{bkp}\nthen the dependence of the dielectric function on\nthe Debye mass may get modified. \nHowever, we assume the same screening mass scale $m_{{}_D}$ which\nemerges in the Debye screened Coulomb potential also appears in the\nnon-perturbative long-distance contribution due to string.\nIn the following section, we take over this assumption to\nanisotropic medium too. However, different scales for the\nCoulomb and linear pieces of the T=0 potential, rather than a single \none, was already employed in Ref.~\\cite{megiasind,megiasprd}.\nMoreover, they developed a theoretical model to include non-perturbative \neffects beyond the deconfinement temperature through dimension-two gluon \ncondensates to calculate the heavy quark free energy.\nInterestingly, their model predicts a duality between the zero temperature \n$Q \\bar Q$ potential and the quark self energy and explains the lattice \ndata well.\n\n\\noindent Finally, one can define a dielectric permittivity\nin one-loop by\n\\begin{eqnarray}\n\\label{eqn2}\n\\epsilon(k)=\\left(1+\\frac{\\Pi_L (0,k,T)}{k^2}\\right)\n\\equiv \\left( 1+ \\frac{m_{{}_D}^2}{k^2} \\right)~.\n\\end{eqnarray}\nAfter substituting the dielectric permittivity in the Fourier \ntransform of Cornell potential (\\ref{eq1}) and then evaluating its \ninverse FT, one obtains the medium modified\npotential in the co-ordinate space~\\cite{prc_vineet,chic_vineet}\n\\begin{eqnarray}\n\\label{defn}\nV(r,T)&=&\\int \\frac{d^3\\mathbf k}{{(2\\pi)}^{3\/2}}\n e^{i\\mathbf{k} \\cdot \\mathbf{r}}~\n\\tilde{V}(k) \\nonumber\\\\\n&=&\\left(\\frac{2\\sigma}{m_D}-\\alpha m_D\\right)\\frac{\\exp{(-\\hat r)}}{\\hat r}\n-\\frac{2\\sigma}{m_D \\hat r}+\\frac{2\\sigma}{m_D}-\\alpha m_D\n\\end{eqnarray}\nwith the dimensionless variable $\\hat{r}=m_{{}_D} r$. The constant terms are introduced  to yield \nthe correct limit of $V(r,T)$ as $T\\rightarrow 0$. Such terms could arise\nnaturally from the basic computations of real time static potential\nin hot QCD~\\cite{Laine2} and also from the real and imaginary time\ncorrelators in a thermal QCD medium~\\cite{beraudo}.\nThe medium modified potential thus obtained has an additional \nlong range Coulomb term with an (reduced) effective charge, in addition\nto the conventional Yukawa term. \n\nIn the small distance limit, $r\\ll 1\/m_{{}_D}$, the above potential reduces to\nthe Cornell potential, {\\it i.e.} $Q \\bar Q$ does not see the medium.\nOn the other hand, in the screening region \n$r\\gg1\/m_{{}_D}$, the potential (\\ref{defn}) reduces to \n\\begin{eqnarray}\n\\label{lrp}\n{V(r,T)}\\sim -\\frac{2\\sigma}{m^2_{{}_D}r}-\\alpha m_{{}_D},\n\\end{eqnarray}\nwhich, apart from a constant term, looks like a Coulomb potential encountered\nin hydrogen-atom problem after identifying the fine structure constant\n$e^2$ with the effective charge $2 \\sigma\/m_{{}_D}^2$. \nThe binding energies and the dissociation temperatures for \nquarkonium states can thus be determined by solving the Schr\\\"odinger\nequation numerically either with the full potential (\\ref{defn}) or \nanalytically with the approximated form (\\ref{lrp}). \nIn addition, one can also exploit the advantage to demonstrate the flavor \ndependence of the dissociation process where the dissociation temperatures \nfor 2-flavor are found to be higher than \nthe 3-flavor case~\\cite{prc_vineet,chic_vineet}. \n\n\n\\subsection{For anisotropic medium {\\texorpdfstring{$(\\xi \\ne 0)$}{xi=0}}}\\label{aniso}\n\\subsubsection{Dielectric permittivity tensor}\nTo study the perturbative potential with an anisotropic parton \ndistribution, consider a hot QCD plasma which, due to expansion \nand finite (momentum) relaxation time, manifests a local anisotropy in\nmomentum space through the distribution function \n\\begin{equation}                                        \nf_{\\rm{aniso}} (\\mathbf{k}) = f_{\\rm{iso}}\\left( \\sqrt{\\mathbf{k}^{2} + \\xi(\\mathbf{k}.\\mathbf{n})^{2}}  \\right) \\quad,\n\\end{equation}                                            \n{\\it i.e.}, $f_{\\rm{aniso}} (\\mathbf{k})$ is obtained from an isotropic distribution $f_{iso} (|\\mathbf{k}|)$ \nby removing particles with a large momentum component along\nthe direction of anisotropy, $\\mathbf{n}$~\\cite{Romatschke}. We shall restrict \nourselves to a plasma close to equilibrium and so that $f_{\\rm{aniso}}(\\mathbf{k})$ \nis either a Bose-Einstein $n_{B} (k)$ or a Fermi-Dirac\n$n_{F} (k)$ distribution function. This may be true because by the \ntime quarkonia have been formed in the plasma medium from the\n$Q \\bar Q$ pairs produced at very early stages of the collision, the system \nmay not be then highly anisotropic \nrather closer to isotropic distribution. In the limit of small anisotropy,\nanisotropy parameter $ \\xi $ is related to the shear \nviscosity-to-entropy density ($\\eta\/s$) through the one-dimensional Navier \nStokes formula by\n\\begin{equation}\n\\xi= \\frac{10}{T \\tau}~\\frac{\\eta}{s},~                 \n\\end{equation}                                   \nwhere $1\/\\tau$ denotes the expansion rate of the fluid element.\nHowever, the degree of momentum-space anisotropy is generically \ndefined by the parameter,\n\\begin{equation}\n\\xi = \\frac{\\langle \\mathbf{k}_{T}^{2}\\rangle}{2\\langle k_{L}^{2}\\rangle}-1~,~\n\\label{anparameter}\n\\end{equation}\nwhere $ {k}_{L}= \\mathbf{k}.\\mathbf{n} $ and ${\\bf k}_T = \n\\mathbf{k}-\\mathbf{n}(\\mathbf{k}.\\mathbf{n}) $ are the components of momentum\nparallel and perpendicular to the direction of \nanisotropy, $\\mathbf{n}$, respectively.\nThe positive and negative values of $ \\xi$ corresponds to the \nsqueezing and the stretching of the distribution function in the direction of \nanisotropy, respectively. In the relativistic nucleus-nucleus \ncollisions, $\\xi$ is however, found to be positive.\nThe calculation of the real part of the potential at finite  anisotropy was \nfirst obtained in Ref. \\cite{Laine2,Romatschke,Dumitru08} and  was later \nextended to calculate the imaginary \npart~\\cite{Brambilla08,beraudo,Laine09,Dumitru09} which is seen as a generic\nfeature of the medium. \nTo study the effect of anisotropy on the in-medium potential, one need \nto calculate first the self-energy in an anisotropic medium. With the specified anisotropic \ndistribution function, we can compute the gluon self-energy \nanalytically~\\cite{Thoma62}. \nWe will restrict our consideration to the spatial part of the self-energy, $\\Pi^{\\mu\\nu} $ \nfor  simplicity and the time-like components can be easily obtained \nby using the symmetry and transversality of the gluon self-energy tensor.\nThe spatial components of the retarded self-energy tensor reads~ \\cite{Dumitru08}\n\\begin{equation}\n\\Pi^{ij}(P)= -g^{2}\\int d^{3}\\mathbf{k}~v^{i}\\frac{\\partial f(\\mathbf{k})}{\\partial \nk^{l}}\\bigg(\\delta^{jl}+\\frac{v^{j}p^{l}}{ P\\cdot V+i\\epsilon}\n\\bigg) \\quad,\n\\label{spatialselfeenrgy}\n\\end{equation}\nwhere $P^{\\mu}\\equiv(p_0,p)$ is the four momentum of the external gluon, \n$p=|{\\bf p}|=$ is the amplitude of the spatial momentum. The four-velocity,\n$V^\\mu$ $(1,{\\bf v}={\\bf k}\/|\\bf k|)$ is a light-like four vector\nwith $v=|{\\bf v}|$ and the partial-derivative, \n$\\partial f \/\\partial k^l$, in terms of new variable $\\tilde{k}$, is given by \n\\begin{equation}\n\\frac{\\partial f(\\mathbf{k})}{\\partial k^{l}}= \\frac{v^{l}+\\xi(\\mathbf{v} .\\mathbf{n})n^{l}}\n{\\sqrt{1+\\xi(\\mathbf{v} .\\mathbf{n})^{2}}}\\frac{\\partial f(\\tilde {k}^{2})}{\\partial\\tilde {k}},\n\\end{equation}\nwhere $\\tilde{k}^{2}= k^{2}(1+\\xi(\\mathbf{v} .\\mathbf{n})^{2})$.  After \nintegrating out over the modified momentum $\\tilde {k}$, the \nself-energy, $\\Pi_{ij}$ is simplified into~\\cite{Dumitru08}\n\\begin{equation}\n\\Pi^{ij}(P)= m_{{}_D}^{2}\\int \\frac{d\\Omega}{4\\pi}v^{i}\\frac{v^{l}+\\xi(\\mathbf{v} .\n\\mathbf{n})n^{l}}{(1+\\xi(\\mathbf{v} .\\mathbf{n})^{2})^{2}}\\bigg(\\delta^{jl}+\\frac{v^{j}p^{l}}{P\\cdot\n V+ i \\epsilon}\\bigg) \\quad,\n\\end{equation}\nwhere the square of the Debye mass is defined by\n\\begin{equation}\nm_{{}_D}^{2}= -\\frac{g^{2}}{2\\pi^{2}}\\int\\limits_{0}^{\\infty} dkk^{2}\\frac{d f_{iso}\n(k^{2})}{dk}.\n\\label{deb}\n\\end{equation}\nUnlike in isotropic medium, the self-energy $\\Pi^{\\rm{ij}}$ shows an \nextra dependence on the preferred anisotropic \ndirection ($\\mathbf{n}$), therefore, it can no longer be decomposed \ninto transverse and longitudinal parts rather it becomes a tensor \n\\cite{Romatschke,Strickland,Strick04} with more basis vectors. \nSo the self-energy tensor \nbe decomposed into four structure functions as~\\cite{Romatschke}\n\\begin{equation}\n\\Pi^{ij}= \\alpha A^{ij}+ \\beta B^{ij} + \\gamma C^{ij} + \\delta D^{ij}~,~\n\\end{equation}\nwhere the coefficients $\\alpha$, $\\beta$, $\\gamma$ and $\\delta$ can be \ndetermined for any value of $\\xi $~\\cite{Romatschke}. \nUsing these structure functions, one can find the gluon propagator in temporal axial gauge as~\\cite{Romatschke}\n\\begin{eqnarray}\n\\Delta_{ij}(\\omega,k)&=&\\frac{(A_{ij}-C_{ij})}{k^{2}-\\omega^{2}+\\alpha}\n+\\frac{(k^{2}- \\omega^{2}+\\alpha +\\gamma)B_{ij}}{(k^{2}-\\omega^{2}+\\alpha +\\gamma)\n(\\beta-\\omega^{2})-k^{2}{\\tilde n}^{2}\\delta^{2}}\\nonumber\\\\\n&&+\\frac{(\\beta-\\omega^{2})C_{ij}-\\delta D_{ij}}{(k^{2}-\\omega^{2}+\\alpha\n +\\gamma)(\\beta-\\omega^{2})-k^{2}{\\tilde n}^{2}\\delta^{2}}\n\\end{eqnarray}\nIn order to see how the anisotropy affects the\nresponse to static electric field, we examine the\npropagator in the  static limit ($\\omega\\rightarrow 0$). Defining the masses\nin the static limit~\\cite{Romatschke}\n\\begin{eqnarray}\nm_{\\alpha}^{2}&=&\\lim_{\\omega \\to 0}\\alpha,~~\nm_{\\beta}^{2}=\\lim_{\\omega \\to 0}-\\frac{k^{2}}{\\omega^{2}}\\beta \\nonumber\\\\\nm_{\\gamma}^{2}&=&\\lim_{\\omega \\to 0}\\gamma,~~\nm_{\\delta}^{2}=\\lim_{\\omega \\to 0}\\frac{\\tilde n k^{2}}{\\omega}{\\rm Im} \\ \\delta \\quad,\n\\label{mminus}\n\\end{eqnarray}\nthe gluon propagator becomes \n\\begin{eqnarray}\n\\label{epsaniso}\n\\lim_{\\omega \\rightarrow 0} \\Delta_{ij}(\\omega,k)=-\\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})\\ k_ik_j}\n{\\omega^2\\left[(k^{2}+ m_{\\alpha}^{2}+m_{\\gamma}^{2})(k^{2}+m_{\\beta}^{2})-\nm_{\\delta}^{4}\\right]}~.\n\\end{eqnarray}\nWe can now factorize the denominator of the gluon propagator as \n\\begin{eqnarray}\n(k^{2}+ m_{\\alpha}^2+m_{\\gamma}^2)(k^2+m_\\beta^2)- \nm_{\\delta}^4=(k^2+m_+^2) (k^2+m_-^2)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n2 m_\\pm^2=M^2 \\pm \\sqrt{M^4-4(m_\\beta^2(m_\\alpha^2+m_\\gamma^2)-m_\\delta^4)}\\ \n,\\ \\ M^2=m_\\alpha^2+m_\\beta^2+m_\\gamma^2~.\n\\end{eqnarray}\nThus the dielectric permittivity for an anisotropic medium in the temporal \naxial gauge can be obtained from the definition~\\cite{kapusta}\n\\begin{eqnarray}\n\\epsilon^{-1}(k)=-\\lim_{\\omega \\to 0} {\\omega^2\\frac{k_ik_j}{k^2}\\Delta_{ij}(\\omega,k)}\n=\\frac{k^{2}(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}\n{(k^2+ m_+^2)(k^{2}+ m_-^2)}~,\n\\label{permittivity}\n\\end{eqnarray}\nIf the anisotropy is small, the pole masses (\\ref{mminus}) can be simplified, \nby retaining only the linear term in $\\xi$, into~\\cite{Romatschke} \n\\begin{eqnarray}\nm_\\alpha^2&=&-\\frac{\\xi}{6}(1+\\cos 2 \\beta_n)~m_{{}_D}^2\\nonumber\\\\\nm_\\beta^2&=&\\left( 1+\\frac{\\xi}{6}(3\\cos 2 \\beta_n-1)\\right)~m_{{}_D}^2\\nonumber\\\\\nm_\\gamma^2&=&\\frac{\\xi}{3} \\sin^2\\beta_n~m_{{}_D}^2\\nonumber\\\\\nm_\\delta^2&=&-\\xi\\frac{\\pi}{4}\\sin \\beta_n \\cos\\beta_n~m_{{}_D}^2\n\\end{eqnarray}\nwhere $\\beta_n$ is the angle between ${\\bf k }$ and ${\\bf n}$ .\nSo the explicit dependencies of $m_\\pm$ on the anisotropy, in the small\n$\\xi$ limit, are given by\n\\begin{eqnarray}\nm_+^{2}&=&\\left(1+\\frac{\\xi}{6}(3\\ \\cos2\\beta_n-1)\\right)m_{{}_D}^{2},\\nonumber\\\\\nm_-^{2}&=&-\\frac{\\xi}{3}\\ \\cos2\\beta_n m_{{}_{D}}^{2}.\n\\label{mplus}\n\\end{eqnarray}\nIn the isotropic limit, all masses become zero except\none ($m_\\alpha^2 = m_\\gamma^2= m_\\delta^2= m_-^2=0, m_+^2=m_{{}_D}^2$),\nwhich is the only pole in the isotropic medium (\\ref{eqn2}).\n\\subsubsection{Medium-modification to heavy quark potential}\nOnce we have obtained the dielectric permittivity in anisotropic \nmedium (\\ref{permittivity}), we substitute it in the Fourier\ntransform (\\ref{eq1}) and then evaluate its inverse Fourier transform\nto obtain the medium modified\npotential in an anisotropic medium:\n\\begin{eqnarray}\nV(\\mathbf{r},\\xi,T)&=&\\frac{1}{{(2\\pi)}^{3\/2}}\n\\int d^{3}\\mathbf{k}~\\tilde{V}(k)~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\\nonumber\\\\\n&=& -\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \n~\\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}{(k^{2}+m_+^2)\n(k^{2}+m_-^{2})}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\n\\nonumber\\\\\n&& -\\frac{4\\sigma}{(2\\pi)^{2}}\\int d^{3}\\mathbf{k}~ \n \\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}{k^{2}(k^{2}+m_+^{2})\n(k^{2}+m_-^{2})}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\n\\label{modipot}\n\\end{eqnarray}\nAfter substituting the pole masses, $m_+$ and $m_-$ (in the small $\\xi$ limit) from (\\ref{mplus}), the potential becomes\n\\begin{eqnarray}\nV({\\bf r},\\xi,T)&=&\\!\\!\\!\\!-\\frac{\\alpha}{2\\pi^{2}}\\int \\frac{d^{3}\\bf{k}  \n~e^{i\\bf{k} \\cdot\\bf{r}}}{\nk^2+m_{{}_D}^2\\left(1+\\frac{\\xi}{6}(3\\cos2\\beta_n-1)\\right)\n} \\nonumber\\\\\n&&\\!\\!\\!\\!-\\frac{4\\sigma}{(2\\pi)^{2}}\n\\int \\frac{d^{3}\\mathbf{k}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}}\n{k^{2}\\left[k^{2}+m_{{}_D}^{2}\\left(1+\\frac{\\xi}{6}(3\\cos2\\beta_n-1)\\right)\\right]}\\nonumber\\\\\n&\\equiv &\\!\\!\\!\\!V_1({\\bf r},\\xi,T)+V_2({\\bf r},\\xi,T)\n\\label{v1}\\ ,\n\\end{eqnarray}\nwhere $V_1 ({\\bf r},\\xi,T)$ and $V_2(\\mathbf{r},\\xi,T)$ are the \nmedium-modified potential corresponding to short-distance Coulombic and \nlong-distance string term, respectively, can be rewritten as \n\\begin{eqnarray}\nV_1({\\bf r},\\xi,T) =-\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \ne^{i\\mathbf{k} \\cdot \\mathbf{r}}\\frac{1}{\\left( k^{2}+{m_{D}}^{2}\\right)}\\left(1+\\frac{\\xi}{6}~\\frac{{m_{D}}^{2}}{\\left( k^{2}+{m_{D}}^{2}\\right)}~(3\\cos2\\beta_n-1)\\right)^{-1}\n\\end{eqnarray}\nExpanding the integrand in terms of the anisotropy parameter ($\\xi$) and \nretaining the term linear in $\\xi$ (weak-anisotropy limit, $\\xi<1$),\n$V_1({\\bf r},\\xi,T)$ can be written as\n\\begin{eqnarray}\nV_1({\\bf r},\\xi,T) &=&-\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \ne^{i\\mathbf{k} \\cdot \\mathbf{r}}\\left[ \\frac{1}{\\left( k^{2}+{m_{D}}^{2}\\right)}-\\frac{\\xi}{6}~\\frac{{m_{D}}^{2}}{\\left( k^{2}+{m_{D}}^{2}\\right)^{2}}~(3\\cos 2\\beta_n-1)\\right]\\nonumber\\\\\n&\\equiv & V_1^{(1)}({ r},\\xi =0,T)+ V_1^{(2)}({\\bf r},\\xi,T)\n\\end{eqnarray}\nwhere $V_1^{(1)}({ r},\\xi=0,T)$ and $V_1^{(2)}({\\bf r},\\xi,T)$ are \nthe isotropic and anisotropic contributions due\nto the medium modification of the Coulomb term, respectively.\nSimilarly $V_2({\\bf r},\\xi,T)$ can be decomposed into isotropic and anisotropic\nparts :\n\\begin{eqnarray}\nV_2({\\bf r},\\xi,T)=&V_2^{(1)}({r},\\xi =0,T)+ V_2^{(2)}({\\bf r},\\xi,T)\n\\end{eqnarray}\nwhere $V_2^{(1)}({r},\\xi=0,T)$ and $V_2^{(2)}({\\bf{r}},\\xi,T)$ are the\nisotropic and anisotropic contributions due\nto the medium modification of the linear term, respectively.\nLet us now calculate them one-by-one. \n\nThe isotropic part, $V_1^{(1)}({ r},\\xi=0,T)$ of\nthe Coulomb term (already calculated in (\\ref{defn}))\nis given by ($\\hat{r}=r~m_D$)\n\\begin{eqnarray}\nV_1^{(1)}({r}, \\xi =0,T)\n=-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}} -\\alpha m_{{}_D}\\ ,\n\\label{col1}\n\\end{eqnarray}\nand the anisotropic part, $V_1^{(2)}({\\bf r},\\xi,T)$ is given by \n\\begin{eqnarray}\nV_1^{(2)}({\\bf r},\\xi,T) = -\\xi~\\frac{\\alpha~ m_{{}_D}^2}{2 \\pi^2}\\int \n\\frac{d^3\\mathbf{k} e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{(k^2+m_{{}_D}^2)^2}\n~\\left(\\frac{2}{3}-\\cos^2\\beta_n\\right). \n\\end{eqnarray}\nOne immediately observes that unlike the isotropic part $V_1^{(1)}(r, \\xi \n=0,T)$, momentum anisotropy ($\\xi \\ne 0$) causes the potential to depend\non angle, in addition to inter-particle distance (r).\nBefore deriving a general angular dependence we first illustrate the two \ncases which are especially interesting to grasp the effect of \nanisotropy on the heavy-quark potential.\nHowever they will be used further to derive the general angular dependence.\nFirst, we consider that $ \\mathbf{r} $ is parallel to the \ndirection of anisotropy, $\\mathbf{n}$, where we have taken the \ndirection of anisotropy $\\mathbf{n}$ along the $z$-axis and \nthe angle between the $r$ and $\\bf k$ is $\\theta$. \nSo the anisotropic part, $ V_1^{(2)}({\\bf r},\\xi,T)$ for the medium\nmodification to the Coulomb term becomes\n\\begin{eqnarray}\n V_1^{(2)}({\\bf r}||{\\bf n},\\xi,T)&=&-\\xi~\\frac{\\alpha m_{{}_D}^2}{\\pi}~\n~\\int \\frac{d^3 \\mathbf{k}~e^{i{\\bf k} \\cdot {\\bf r}}}{(k^2+m_{{}_D}^2)^2}~ \n\\left(\\frac{2}{3}-\\cos^2\\theta\\right).\n\\end{eqnarray}\nAfter the angular integration, it becomes simplified as\n\\begin{eqnarray}\n V_1^{(2)}({\\bf r}||{\\bf n},\\xi,T)\n&=& \\xi~V_1^{(1)}({ r},\\xi=0,T)\\left(2~\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\n\\frac{2}{\\hat{r}}-\\frac{\\hat{r}}{6}-1\\right).\n\\label{col2}\n\\end{eqnarray}\nThus, the complete in-medium modification \nto the Coulomb term (Eqs. \\ref{col1} and \\ref{col2}), for the parallel alignment \nbecomes\n\\begin{eqnarray}\n V_1({\\bf r}||{\\bf n}, \\xi,T) &=&V_1^{(1)}({ r},\\xi=0,T)+ \nV_1^{(2)} ({\\bf r}||{\\bf n},\\xi,T) \\nonumber\\\\\n&=&-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}}-\\alpha m_{{}_D}-\\xi~\n\\frac{\\alpha m_{{}_D}}{\\hat{r}}~e^{-\\hat{r}}\\left(2\n\\frac{e^{\\hat r}-1}{{\\hat r}^2}-\\frac{2}{\\hat r}-\\frac{\\hat r}{6}-1\\right).\n\\label{col}\n\\end{eqnarray}\nNext we consider the other scenario, i.e.  when $ \\mathbf{r} $ is \ntransverse to the direction of anisotropy, ${\\bf n}$ \nwhere we take {\\bf r} along the $z$-axis and {\\bf n} lying in the $x$-$y$ plane, so\n$\\phi$ is the azimuthal angle and $\\phi_n$ is the angle between \n$\\mathbf{n}$ with $x$-axis.  Then $V_1^{(2)}({\\bf r} \\perp {\\bf n},\\xi,T)$ \nbecomes\n\\begin{eqnarray}\nV_1^{(2)}({\\bf r \\perp n},\\xi,T)\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!-\\xi\n~\\frac{\\alpha m_{{}_D}^2}{2\\pi^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i {\\bf k} \\cdot {\\bf r}} }{(k^2+m_{{}_D}^2)^2}~\n\\left(\\frac{2}{3}-\\cos^2\n\\left(\\phi-\\phi_n\\right)\\sin^2\\theta\\right).\n\\end{eqnarray}\nAfter the angular integration (exploiting the cylindrical symmetry), it \nis simplified into\n\\begin{eqnarray}\nV_1^{(2)}({\\bf r \\perp n},\\xi,T)=\\xi V_1^{(1)}({ r},\\xi=0,T)\\left(\\frac{1-e^{\\hat{r}}}{{\\hat{r}}^2}\n+\\frac{1}{\\hat{r}} +\\frac{\\hat{r}}{3}+\\frac{1}{2}\\right)\n\\label{colperp}\n\\end{eqnarray}\nThus, the complete in-medium modification \nto the Coulomb term (Eqs. \\ref{col1} and \\ref{colperp}), for the transverse \nalignment becomes\n\\begin{eqnarray}\nV_1({\\bf r\\perp n},\\xi, T) &=& V_1^{(1)}({ r},\\xi=0,T)\n        + V_1^{(2)}({\\bf r\\perp n},\\xi,T) \\nonumber\\\\\n      &=&-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}}-\\alpha m_{{}_D}\n       +\\xi~\\frac{\\alpha m_{{}_D}}{\\hat{r}} e^{-\\hat{r}}\n       \\left(\\frac{e^{\\hat{r}}-1}{{\\hat{r}}^2}-\\frac{1}{\\hat{r}}\n      -\\frac{\\hat{r}}{3} -\\frac{1}{2}\\right).\n\\label{couperp}\n\\end{eqnarray}\nNext we calculate the in-medium-modification to the linear term, \n$V_2({\\bf r},\\xi,T)$ where the isotropic part (from (\\ref{defn})) is given by \n\\begin{equation}\nV_2^{(1)}({r},\\xi =0,T)=\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\ne^{-\\hat{r}}-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\n\\label{lin1}\n\\end{equation}\nand the anisotropic contribution, when $ \\mathbf{r} $ is parallel to the \ndirection of anisotropy $\\bf n$, is given by\n\\begin{eqnarray}\nV_2^{(2)}({\\bf r || n},\\xi,T) &=&-\\xi~\\frac{4\\sigma}{(2\\pi)^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{k^2~(k^2+m_{{}_D}^2)^2}\n~\\left(\\frac{2}{3}-\\cos^2\\beta_n\\right) .\n\\end{eqnarray}\nAfter the angular integration, it becomes\n\\begin{eqnarray}\nV_2^{(2)}({\\bf r || n},\\xi,T)\n&=& -\\xi \\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(2\\frac{(1-e^{\\hat r})}{\\hat r^2}+\n\\frac{e^{\\hat r}+2}{3} +\\frac{2}{\\hat r}+\\frac{\\hat {r}}{12}\\right)\n\\label{lin2}\n\\end{eqnarray}\nThus, the complete in-medium modification to the linear term (Eqs. \\ref{lin1} \nand \\ref{lin2}), for $ \\mathbf{r} \\ || \\mathbf{n} $ becomes\n\\begin{eqnarray}\nV_2({\\bf r|| n}, \\xi, T) &=&\nV_2^{(1)}({r},\\xi=0,T)+ V_2^{(2)}({\\bf r|| n},\\xi,T)\n\\nonumber\\\\\n      &=&-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n   + \\frac{2\\sigma}{m_{{}_D} \\hat r} e^{-\\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\\nonumber\\\\\n     &+&\\xi~\\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(2~\n\\frac{e^{\\hat{r}}-1}{{\\hat{r}}^2}-\\frac{e^{\\hat{r}}+2}{3} -\\frac{2}{\\hat{r}}\n-\\frac{\\hat{r}}{12}\\right)\n\\label{lin}\n\\end{eqnarray}\nOn the other hand, when $ \\mathbf{r} $ is transverse to the direction of anisotropy,\n$\\mathbf{n}$, the anisotropic contribution to the linear term becomes\n\\begin{eqnarray}\nV_2^{(2)}({\\bf{r} \\perp\\bf{n}},\\xi,T) &=&-\\xi~\\frac{4\\sigma}{(2\\pi)^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{k^2~(k^2+m_{{}_D}^2)^2}\n~ \\left(\\frac{2}{3}-\\cos^2 \\left(\\phi-\\phi_n\\right)\\sin^2\\theta\\right), \n\\end{eqnarray}\nwhich is simplified into\n\\begin{eqnarray}\nV_2^{(2)}({\\bf{r} \\perp\\bf{n}},\\xi,T) &=&\n-\\xi ~\\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right) ,\n\\label{linp}\n\\end{eqnarray}\nafter the angular integration. Thus, the complete in-medium modification to the linear term \n(Eqs. \\ref{lin1} and \\ref{linp})\nfor $ \\mathbf{r} \\perp \\mathbf{n} $ becomes\n\\begin{eqnarray}\nV_2({\\bf{r}\\perp\\bf{n}}, \\xi, T) &=&\nV_2^{(1)}({r},\\xi=0,T)+ V_2^{(2)}({\\bf r\\perp\\bf{n}},\\xi,T)\n\\nonumber\\\\\n&=&-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n+ \\frac{2\\sigma}{m_{{}_D} \\hat{r}} e^{-\\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\\nonumber\\\\\n&-&\\xi ~\\frac{4\\sigma}{m_{{}_D}^2 r}{e^{-m_{{}_D} r}}\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right).\n\\label{linperp}\n\\end{eqnarray}\n\\par Therefore, the full medium-modified potential, consisting of Coulomb\nand linear term (Eqs. \\ref{col} and \\ref{lin}, respectively), for the\nquark pairs aligned to the direction of anisotropy, yields as \n\\begin{eqnarray}\nV({\\bf{r} \\parallel \\bf{n}},\\xi,T)\n&=& \\left(\\frac{2\\sigma}{m_{{}_D}}-\\alpha m_{{}_D} \\right)\\frac{e^{-\\hat r}}{\\hat{r}}\n-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&+& \\xi~\\left[ \\frac{4\\sigma}{m_{{}_D} \\hat{r}}~e^{-\\hat r} \\left( {} 2~\\frac{e^{\\hat r}-1}\n{\\hat r^2}-\\frac{e^{\\hat r}+2}{3} -\\frac{2}{\\hat r}-\\frac{\\hat r}{12} \\right) \\right. \\nonumber\\\\\n&-&\\left. \\frac{\\alpha m_{{}_D} }{\\hat{r}} e^{-\\hat r}\\left( {} 2~\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n-\\frac{2}{\\hat r}-\\frac{\\hat r}{6} -1\\right) \\right] \\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (r,T)+ V_{\\rm{aniso}}^\\parallel (r,\\xi,T).\n\\label{fullpotpara}\n\\end{eqnarray}\nIn the short distance limit ($r \\ll 1\/m_{{}_D}$), the potential \nreduces to the vacuum potential (Cornell) for $\\xi=0$, {\\it i.e.} \n$Q\\bar Q$ pairs are not affected by \nthe medium. On the other hand, in the long-distance limit\n($r\\gg 1\/m_{{}_D}$) (where the screening occurs), we can \nneglect the Yukawa term and for large values of temperatures, the product\n$\\alpha m_{{}_D}$ will be much greater than $2\\sigma\/m_{{}_D}$. Thus\nthe potential is simplified into the following form:\n\\begin{eqnarray}\nV({\\mathbf r} \\parallel {\\mathbf n},\\xi,T) &\\stackrel{\\hat{r}\\gg 1}{\\simeq} & -\\frac{2\\sigma}{m_{{}_D}^{2}r} - \n\\alpha m_{{}_D} -\\frac{4\\xi}{6}~\\left(\\frac{2\\sigma}{m_{{}_D}^{2}r} \\right) \\nonumber\\\\\n&\\stackrel{\\hat{r}\\gg 1}{\\equiv} & V_{\\rm{iso}} (\\hat{r} \\gg 1,T)+ V_{\\rm{aniso}}^\\parallel\n(\\hat{r} \\gg 1,\\xi,T),\n\\label{paral}\n\\end{eqnarray}\nwhich clearly shows that the potential for $Q \\bar Q$ pairs aligned in the \ndirection of anisotropy gets screened less, {\\em i.e.} becomes stronger \ncompared to isotropic medium (\\ref{lrp}).\n\n\\noindent On the other hand, when the quark pairs are aligned transverse to the \ndirection of anisotropy ($ \\mathbf{r}  \\perp \\mathbf{n} $), \nthe medium modification to the Coulombic and linear terms together \n(Eqs. \\ref{couperp} and \\ref{linperp}, respectively) gives rise to \nthe following form:\n\\begin{eqnarray}\nV({\\bf r\\perp {\\bf n}},\\xi,T)&=& \n\\left(\\frac{2\\sigma}{m_{{}_D} }-\\alpha m_{{}_D} \\right) \n\\frac{e^{-\\hat r}}{\\hat{r}}-\\frac{2\\sigma}{m_{{}_D}\\hat{r}}  \n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&-&\\xi~\\left[ \\frac{4\\sigma}{m_{{}_D} \\hat{r}}~{e^{-\\hat{r}}} \n\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right) \\right.\n\\nonumber\\\\\n&-& \\left. \\frac{\\alpha m_{{}_D}}{\\hat{r}} e^{-\\hat{r}}\n\\left(\\frac{e^{\\hat r}-1}{\\hat r^2}-\\frac{1}{\\hat r}-\\frac{1}{2}\n-\\frac{\\hat r}{3}\\right) \\right] \\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (r,T)+ V_{\\rm{aniso}}^\\perp (r,\\xi,T),\n\\label{fullpotperp}\n\\end{eqnarray}\nSimilarly, $Q \\bar Q$ pair does not see the medium, in the short-distance \nlimit whereas in the long-distance limit, the \npotential is simplified into a Coulombic form with\na dynamically screened color charge ($2\\sigma\/m_D^2$):\n\\begin{eqnarray}\nV({\\bf r} \\perp {\\bf n},\\xi,T)&\\stackrel{\\hat{r}\\gg1}{\\simeq}&-\\frac{2\\sigma}{m_{D}^{2}r}\n-\\alpha m_{D}-\\frac{\\xi}{6} \\left(\\frac{2\\sigma}{m_{D}^{2}r}\\right)\\nonumber\\\\\n&\\stackrel{\\hat{r}\\gg 1}{\\equiv} & V_{\\rm{iso}} (\\hat{r}\\gg 1,T)+ V_{\\rm{aniso}}^\\perp \n(\\hat{r}\\gg 1,\\xi,T),\n\\label{perp} \n\\end{eqnarray}\nwhich again shows that the potential for the transverse alignment is still\nstronger than in isotropic medium but less stronger than the former.\n\n\\par Until now we have demonstrated the effects of momentum anisotropy on\nthe heavy-quark interaction for the special cases {\\em viz} the inter-particle \nseparation (r) may be parallel or perpendicular to the direction of \nanisotropy (n). We would now like now to derive a potential which depends \non both the inter-particle separation (r) and the\nangle ($\\theta_n$) between ${\\bf r}$ and $n$, explicitly.\n\nLet us assume that ${\\bf r}$  is parallel to the z component of ${\\bf k}$\nand the direction of anisotropy $ {\\bf n} $ lies in the $ y$-$z$\nplane (cylindrical symmetry) in the momentum space.\nWe may further assume that given the weak anisotropy, the potential in \nthe anisotropic medium represents a perturbation to the central potential \nin the isotropic medium as : \n\\begin{eqnarray}\nV(r,\\theta_n, T)&=&V(r,T)+ V_{\\rm{tensor}}(r,\\theta_n,T)\\\\\n&=& V_{\\rm{iso}}(r,T)+\\xi F(r,\\theta_n,T).\n\\label{potgen}\n\\end{eqnarray}\nwhere the tensorial part $V_{\\rm{tensor}}(r,\\theta_n,T)$ represents a small\nperturbation to the central one $V(r;T)$ and $\\xi$ is the strength of \nnon-central component of the potential. The function $F(r,\\theta_n,T)$ can \nbe expanded as\n\\begin{equation}\nF(r,\\theta_n, T)=f_0(r,T)+f_1(r,T)\\cos 2\\theta_n~.\n\\end{equation}\nThe potentials for the angles $\\theta_n=0$ and $\\theta_n=\\pi\/2$ help us to \ndetermine the functions $ f_0(r,T) $ \nand $f_1(r,T)$ in terms\\footnote{The variable\n$\\hat{r}$ should not be confused with the usual notation of unit vector in\nthe coordinate system.} of $\\hat{r}$ (=$rm_D$) as\n\\begin{eqnarray}\nf_0(\\hat{r},T)&=&\\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}} \n\\left(\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{5 e^{\\hat{r}}}{12}-\n\\frac{1}{\\hat{r}}+\\frac{\\hat{r}}{12} -\\frac{1}{12}\\right)\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}} {2}\\frac{e^{-\\hat{r}}}{\\hat{r}}\n\\left(\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{1} {\\hat{r}}+\n\\frac{\\hat{r}}{6}-\\frac{1}{2}\\right)\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nf_1(\\hat{r},T)&=&\\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}}\\left(\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{e^{\\hat{r}}}{4}-\n\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{4}-\\frac{5}{4}\\right)\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}}{2} \\frac{e^{-\\hat{r}}}{\\hat{r}}\\left(\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{2}-\n\\frac{3}{2}\\right)\n\\end{eqnarray}\nSo after substituting $f_0$ and $f_1$ into the Eq.(\\ref{potgen}), we obtain\nthe complete angular dependence of the potential in \nthe limit of weak anisotropy ($\\xi \\ll 1$) :\n\\begin{eqnarray}\nV(r,\\theta_n,T) &=& \\left(\\frac{2\\sigma}{m_{{}_D} }-\\alpha m_{{}_D} \\right) \n\\frac{e^{-\\hat r}}{\\hat{r}}-\\frac{2\\sigma}{m_{{}_D}\\hat{r}}  \n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&+&  \\xi \\left( \\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}} \n\\left[\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{5 e^{\\hat{r}}}{12}-\n\\frac{1}{\\hat{r}}+\\frac{\\hat{r}}{12} -\\frac{1}{12}\\right] \\right. \\nonumber\\\\\n&-& \\left. \\frac{\\alpha m_{D}} {2}\\frac{e^{-\\hat{r}}}{\\hat{r}}\n\\left[\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{1} {\\hat{r}}+\n\\frac{\\hat{r}}{6}-\\frac{1}{2}\\right] \\right. \\nonumber\\\\\n&+ & \\left( \\left. \\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}}\\left[\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{e^{\\hat{r}}}{4}-\n\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{4}-\\frac{5}{4}\\right] \\right.\\right.\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}}{2} \\frac{e^{-\\hat{r}}}{\\hat{r}}\\left[ \\left.\\left.\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{2}-\n\\frac{3}{2}\\right] \\right) \\cos~2 \\theta_n \\right) \\nonumber\\\\\n&=& V(r,T) + V_{\\rm{tensor}}(r,\\theta_n,T) \n\\label{fullp}\n\\end{eqnarray}\nThus the anisotropy in the momentum space introduces an angular \n($\\theta_n$) dependence, in addition to the inter-particle separation ($r$),\nto the potential in the coordinate space which was earlier only $r$-dependent \nin the isotropic medium. We can now identify the $\\xi$-independent \nterm with $V(r,T)$ in (50) which depends only on the separation ($r$) distance \nand the $\\xi$-dependent term with the tensorial component \n$V_{\\rm{tensor}}(r,\\theta_n,T)$ in (50) which depends on both \n$r$ and $\\theta_n$. Thus the full potential in an anisotropic \nmedium, $V(r, \\theta_n,T)$ needs to be solved by the three-dimensional \nSchr\\\"odinger equation. Since we are restricted in the small $\\xi$ limit, so \nthe tensorial component $V_{\\rm{tensor}}(r,\\theta_n,T)$ is much smaller than\nthe (isotropic) central component $V(r,T)$ and hence may be treated as the \nperturbation by a first-order perturbation theory in quantum mechanics.\nHowever, the isotropic component may be solved numerically by the \none-dimensional (radial) Schr\\\"odinger equation.\n\nThe heavy quark interaction at short and intermediate distances\n($rm_D \\le1$) are important for the understanding of in-medium\nmodification of heavy quark bound states and the large distance\nproperty ($rm_D>1$) helps to understand the bulk properties of\nQGP phase which also affect the in-medium properties\nof the quarkonium states. So we wish to \nto see how the potential in anisotropic medium behaves in these\n(short, intermediate and long) limiting cases.\nIn the short-distance limit, the vacuum contribution \ndominates over the medium contribution and this is exactly happens \nhere \n\\begin{eqnarray}\nV(r,\\theta_n,T)\\stackrel{\\hat{r}\\ll 1}{\\simeq} \\sigma r - \\frac{\\alpha}{r}\n\\label{vprime}\n\\end{eqnarray}                                        \nfor $\\xi=0$. On the other hand, in the long-distance limit ($\\hat{r}\\gg 1 $), the \npotential is reduced to a long-range Coulombic interaction after\nidentifying the factor $2\\sigma\/m_D^2$ with the coupling ($g_s^2$)\nof the interaction\n\\begin{eqnarray}\n\\label{largp}\nV(r,\\theta_n,T) &\\stackrel{\\hat{r}\\gg1}{\\simeq}& -\\frac{2\\sigma}{m^2_{{}_D}r}-\\alpha m_{{}_D}\n-\\frac{5\\xi}{12}~\\frac{2\\sigma}{m^2_{{}_D}r} \\left(1+\\frac{3}{5}\\cos 2\\theta_n \\right)\\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (\\hat{r} \\gg 1,T)+ V_{\\rm{tensor}} (\\hat{r} \\gg 1~.\n\\theta_n,T)~.\n\\end{eqnarray}\nSince the resulting potential is Coulombic plus a\nsubleading anisotropic contribution, it then has to satisfy\nthe condition: $a_0 m_D\\gg 1$, where $a_0$ is the Bohr radius and $m_D$ is\nthe Debye mass. Since the Bohr radius $a_0$ is proportional to \n$m_D^2\/(m_Q \\sigma)$,\nthe above condition for the long-distance limit implies that \n$m_D^3\/(m_Q \\sigma)$ \nshould be greater than 1. Thus this inequality results in a condition on the\nDebye mass and hence on the temperature. It is seen that the above condition \nis satisfied for the temperatures above the critical temperature ($>T_c$)\nfor the charmonium states and above 1.6$T_c$ \nfor the bottomonium states. The temperature ranges ($T_c$ and \n1.6$T_c$) for $c \\bar c$ and $b\\bar b$ states above which \nthe effective potential looks Coulombic are\nsmaller than their respective dissociation temperatures and thus seems \njustified to approximate the potential in the long-distance limit. In the intermediate distance ($rm_D \\simeq 1$) scale, the \ninteraction becomes complicated and thus the potential\ndoes not look simpler in contrast to the asymptotic limits, so\nthis limit needs to be dealt numerically with the full potential in\na Schr\\\"odinger equation.\n\n\\par We have thus noticed overall that in the short \ndistance limit, the potential have not been affected in the isotropic limit.\nOn the contrary, in the long-distance limit, the momentum anisotropy \ntranspires an angular dependence in the potential \nand gives rise a characteristic angular ($\\theta_n$) dependence \nbetween the relative \nseparation ($\\mathbf{r}$) and the direction of anisotropy ($\\mathbf{n}$). \nAs a corollary, the quark pairs aligned \nalong the direction of anisotropy feel more attraction than the transverse \nalignment because the inter-quark potential along\nthe direction of anisotropy is screened less than the transverse alignment.\nHowever, the potential in the anisotropic medium is always stronger than in \nisotropic medium. \n\n\nTo see the effects of anisotropy, we have shown the potentials \nfor $Q \\bar Q$ pairs in an anisotropic medium in Figures 1 and 2,\nfor $\\theta_n=0$ (parallel) and $\\theta_n=\\pi\/2$\n(perpendicular), respectively. The immediate observation\ncommon to all figures is that the inter-quark potential in anisotropic\nmedium is always more attractive than in isotropic medium. \nThis can be understood physically: In the small \nanisotropic limit, the anisotropic distribution function may be \nobtained from an isotropic distribution $f_{iso} (|\\mathbf{k}|)$\nby removing particles with a large momentum component along\n$\\mathbf{n}$ {\\it i.e.}\n$f_{iso}(\\sqrt{\\mathbf{k}^{2} + \\xi(\\mathbf{k}.\\mathbf{n})^{2}}$.\nThis transpires in the reduction of the number of partons (around\na static test heavy quark) than in isotropic medium {\\it i.e.} $n_{\\rm{aniso}} (\\xi)= \nn_{\\rm{iso}}\/\\sqrt{1+\\xi}$. Therefore, the (effective) Debye mass \nalways becomes smaller and results in less screening of the potential than \nin isotropic medium.\n\\begin{figure}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{para.eps}\n\\label{para_plot_a}}\n\\end{subfigure}\n \\hspace{-5mm}\n\\begin{subfigure}{\n \\includegraphics[width=7.9cm,height=8.5cm]{paraz1.eps}\n\\label{para_plot_b}}\n\\end{subfigure}\n\\caption{\\footnotesize The left panel represents the potential \ndivided by $(g^2 C_F m_{{}_D})$ and the right panel represents the \ncontribution of Coulomb, string  and both together as a function of \n$\\hat{r}$ (= $rm_{{}_D} $) for quark pairs parallel to the \ndirection of anisotropy, $\\mathbf{n}$.} \n\\end{figure}\n\n\n\\begin{figure}\n\\vspace{0mm}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{perp.eps}\n\\label{perp_plot_a}}\n\\end{subfigure}\n\\hspace{-5mm}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{perpz1.eps}\n\\label{perp_plot_b}}\n\\end{subfigure}\n\\caption{\\footnotesize The notations are same as in Figure 1\nbut for quark pairs perpendicular to the \ndirection of anisotropy, $\\mathbf{n}$.} \n\\vspace{0mm}\n\\end{figure}\nThe second observation is that the quark pairs aligned along ($\\theta_n=0$) the\ndirection of anisotropy are stronger than aligned perpendicular \n($\\theta_n=\\pi\/2$) to the direction of anisotropy because for the parallel alignment, the component of momentum to be removed is higher\nthan the transverse alignment so the distribution function \nfor the parallel alignment case contributing to the Debye mass is\nsmaller than the transverse alignment.\nHence the potential for parallel case will be screened less compared to \nthe transverse case. However the difference between the two scenario \nwill be not much different\nbecause the contributions to the Debye mass from the partons having higher \nmomenta are very small.\n \nTo understand the effect of linear term on the medium modified \npotential quantitatively, in addition to the Coulomb term, we have \nplotted separately the medium modifications to the linear term, \nthe Coulomb term and their sum in the right panels of \nFigure 1 and 2, for parallel and transverse case, respectively.\nMedium modification to the Cornell potential contains two parts: one is due to\nthe medium modifications of linear term ($\\sigma r$) and the other one is due \nto the medium modifications of\nCoulomb term. As usually done in the literature, medium \nmodification to the linear term does not arise because\nthe string tension was assumed to be \nzero~\\cite{mocsyprd,Satz,shro,Alb05} \nat or beyond deconfinement temperature~\\cite{Lusch}.\nSince string tension is found to be nonzero at $T_c$ rather it \napproaches zero much beyond $T_c$~\\cite{string1,string2,string3} and hence\nthe medium modification to the linear term may be\nnon-zero contribution to the potential even at temperatures beyond $T_c$, although it is very small. \nIn isotropic medium, medium modification to the linear term\nremains  positive up to 2-3 $T_c$, making the potential less attractive \ncompared to $T=0$. On contrast, in anisotropic case medium \nmodification to the linear term becomes negative\nand the overall full potential becomes more attractive.\n\nAs mentioned earlier we used the same screening scale for both the linear\nand Coulombic terms in our calculation which does not look plausible.\nIt would thus be interesting to see the effects of different scales for the \nCoulomb and linear pieces of the T=0 potential~\\cite{megiasind,megiasprd}.\nTo illustrate it graphically, we have compared our results with their results\n(in Figure 3) for the isotropic case. The difference in the large\ndistance limit arises due to the difference in the potential\nat infinity (-$\\sigma\/m_D$) so the \npotential in Ref.\\cite{megiasprd} is more attractive than our potential.\n\\begin{figure}\n\\vspace{0mm}\n\\includegraphics[width=7.9cm,height=8.5cm]{pot_emegias_our.eps} \n\\caption{\\footnotesize The dotted-circle represents the results from \nMegias et al. \\cite{megiasprd} where\ndifferent scales was used for linear and Coulomb terms separately whereas\nthe solid line represents our work.}\n\\vspace{0mm}\n\\end{figure}\n\n\\section{Properties of Quarkonium in an Anisotropic Medium}\\label{prop_q}\n\\subsection{Binding energy}\nTo understand the in-medium properties of the quarkonium states,\nwe need to model the heavy quark potential as a function of\ntemperature and solve the resulting Schr\\\"{o}dinger equation.\nThe potential thus obtained in anisotropic medium (\\ref{fullp}), \nin contrast to the (spherically symmetric) potential in isotropic medium,\nis non-spherical and so one cannot simply obtain the energy eigen values \nby solving the radial part of the Schr\\\"{o}dinger equation only because \nthe radial part is no longer sufficient due to \nthe angular dependence in the potential. Other way to understand\nis that because of the anisotropic\nscreening scale, the wave functions are no longer radially symmetric for\n$\\xi \\ne 0$.  So one has to solve the potential in anisotropic medium\nthrough the Schr\\\"odinger equation in three dimension. \nHowever, we have seen in the potential (55) that in the small $\\xi$-limit, \nthe spherically non-symmetric component\n$V_{\\rm{tensor}}(r,\\theta_n,T)$ is much smaller in comparison to spherically \nsymmetric (isotropic) component $V(r,T)$ and thus\ncan be treated as perturbation. This can be understood physically:\nThe tensorial (non-sphericity) nature\nof the potential in the co-ordinate space is arisen due to anisotropy \nin the momentum space. However, we are restricted to a plasma which\nis very much close to equilibrium because\nby the time quarkonium states are formed in the plasma\naround (1-2)$T_c$, the plasma becomes almost \nisotropized. Thus  this weak (momentum) anisotropy\n($\\xi \\ll 1$) transpires feeble angular dependence in the potential\nso the potential will be spherically abundant\nwith a tiny non-spherical component. So we could \ntreat the anisotropic component through the perturbation theory in quantum \nmechanics and the isotropic part should be\nhandled numerically by the one-dimensional radial\nSchr\\\"{o}dinger equation.\n\nThere are some numerical methods to solve the Schr\\\"odinger equation\neither in partial differential form (time-dependent) or eigen value form\n(time-independent\/stationary) by the\nfinite difference time domain method (FDTD) or matrix method, respectively.\nHowever, we choose the matrix method to solve the\nstationary Schr\\\"odinger equation with the isotropic part of the potential\n(55) in anisotropic medium. In this method, the Schr\\\"odinger equation\ncan be cast in a matrix form through a discrete basis, instead\nof the continuous real-space position basis spanned by the states\n$|\\overrightarrow{x}\\rangle$. Here the confining potential V is subdivided\ninto N discrete wells with potentials $V_{1},V_{2},...,V_{N+2}$ such that\nfor $i^{\\rm{th}}$ boundary potential, $V=V_{i}$ for $x_{i-1} < x < x_{i};~i=2, 3,..\n.,(N+1)$. Therefore for the existence of a bound state, there\nmust be exponentially decaying wave function\nin the region $x > x_{N+1}$ as $x \\rightarrow  \\infty $ and\nhas the form:\n\\begin{equation}\n\\Psi_{N+2}(x)=P_{{}_E} \\exp[-\\gamma_{{}_{N+2}}(x-x_{N+1})]+ \nQ_{{}_E} \\exp [\\gamma_{{}_{N+2}}(x-x_{N+1})] , \n\\end{equation}\nwhere, $P_{{}_E}= \\frac{1}{2}(A_{N+2}- B_{N+2})$,\n$Q_{{}_E}= \\frac{1}{2}(A_{N+2}+ B_{N+2}) $ and,\n$ \\gamma_{{}_{N+2}} = \\sqrt{2 \\mu(V_{N+2}-E)}$. The eigenvalues\ncan be obtained by identifying the zeros of $Q_{E} $.\n\n\nTherefore, the corrected energy eigen value comes\nfrom the solution of Schr\\\"odinger equation\nof the isotropic component $V_{\\rm{iso}}(r,T)$, using the \nabovementioned matrix method\nplus the first-order perturbation due to the anisotropic \ncomponent $V_{\\rm{aniso}}(r,\\theta;\\xi,T)$ (55) through\nthe quantum mechanical perturbation theory.\nThe variations of the binding energies with the \ntemperature are shown in figure~(\\ref{bind_jsi})\nfor $J\/\\psi$ and $\\Upsilon$ \nfor different values of anisotropy parameter $\\xi$,\nto see the effect of anisotropy on the binding energies compared to\nthe isotropic case.  \n\n\\begin{figure*}\n\\begin{subfigure}{\n\\includegraphics[trim = 1mm 1mm 1mm 1mm, clip,height=7cm,width=7cm]{charm.eps}\n}\n\\end{subfigure}\n\\hspace{5mm}\n\\begin{subfigure}{\n\\includegraphics[trim = 1mm 1mm 1mm 1mm, clip,height=7cm,width=7cm]{bot.eps}\n}\n\\end{subfigure}\n\\vspace{0mm}\n\\caption{\\footnotesize Variation of $J\/\\psi$ and $\\Upsilon$ binding \nenergy (in $GeV$) with the temperature in an anisotropic hot QCD medium.}\n\\label{bind_jsi}\n\\end{figure*}\n\nThere are mainly two observations: First, as the anisotropy increases, the binding of \n$Q \\bar Q$ pairs get\nstronger with respect to their isotropic counterpart because \nthe potential becomes deeper with the increase of anisotropy due to\nweaker screening. It seems \nthat the (effective) Debye mass $m_D(\\xi,T)$ in an anisotropic\nmedium is always smaller than in an isotropic medium. As a result\nthe screening of the Coulomb and string contribution are \nless accentuated and hence the quarkonium states \nbecome more stronger than in an isotropic medium.\nHowever, the effects of anisotropy on the \nexcited states are not \nso pronounced compared to the ground states\nbecause they are generically weakly bound.\nSecondly, there is\na strong decreasing trend with the temperature.\nThis is due to the fact that the screening becomes always stronger with the increase of\ntemperature, so the potential becomes weaker compared to $T=0$ and\nresults in early dissolution of quarkonia in the medium.\nOur results on the temperature dependence of the binding energies show\nan agreement with the similar variations in other\ncalculations~\\cite{Dumitru09}.\n\nIn our calculation, we use the Debye mass \n($m_{{}_D}^{\\rm L}=1.4 m^{\\rm LO}_{{}_D}$) obtained by fitting\nthe (color-singlet) free energy in lattice QCD~\\cite{mocsyprl} \nwhere both one and two-loop expression \n~\\cite{shro,shaung,ijmp} for coupling have been used to explore the\neffects of running coupling on the dissociation process.\n\nThus the study of the temperature dependence of the binding \nenergies are poised\nto provide a wealth of information about the dissociation\npattern of quarkonium states in an anisotropic thermal medium that can be\nused to determine the dissociation temperatures of different states\nin the next Section.\n\n\\subsection{Dissociation temperatures for heavy quarkonia}\\label{diss}\nDissociation of a two-body bound state in an thermal medium can be \nunderstood qualitatively: When the binding energy of a resonance state\ndrops below the mean thermal energy of a parton, \nthe state becomes feebly bound. The thermal\nfluctuation then can easily dissociate by exciting\nthem into the continuum.\nThe spectral function technique in potential models defines the dissociation\ntemperature as the temperature above which the quarkonium spectral\nfunction shows no resonance-like structures but the widths shown \nin spectral functions from current potential model calculations\nare not physical. The broadening of states with the increase in \ntemperature is not included in any of these models. \nIn Ref.\\cite{mocsyprl}, the authors\nargued that one need not to reach the binding energy ($E_{\\rm{bin}}$) to\nbe zero for the dissociation rather a weaker condition $E_{\\rm{bin}}<T$\ncauses a state weakly bound. In fact, when $E_{\\rm{bin}} \\simeq T$, the\nresonances have been broadened due to direct thermal activation, \nso the dissociation of the bound states may be expected to occur\nroughly around $E_{\\rm{bin}} \\simeq T$.\n\nUsing the binding energies calculated earlier in Sec.3.1, \nthe dissociation temperatures ($T_D$) (shown in Table 1)\nare found minimum for the isotropic case and increase\nwith the increase of anisotropy ($\\xi>0$) {\\em viz.} $J\/\\psi$ \nis dissociated at $1.38~T_c$ in an isotropic medium while in an \nanisotropic medium with the anisotropies $\\xi$=0.3 and 0.6, they will \nsurvive higher \ntemperatures, $1.41~T_c$ and $1.43~T_c$, respectively. Similarly \nthe dissociation temperatures of \n$\\Upsilon$ for $\\xi$=0.3 and 0.6 are $1.71~T_c$\nand $1.72~T_c$, respectively, corresponding to the value \n($1.70~T_c$) in an isotropic medium. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nState &$\\xi=0.0$ & $\\xi=0.3$ & $\\xi=0.6$\\\\\n\\hline\\hline\n$J \/ \\psi$ & 1.38& 1.41& 1.43  \\\\\n\\hline\n$\\Upsilon$ & 1.70 & 1.71 & 1.72 \\\\\n\\hline\n\\end{tabular} \n\\caption{\\footnotesize Dissociation temperatures ($T_D$)\nfor the quarkonium states with one-loop QCD coupling}\n\\label{tdpara1l}\n\\end{table}\n\\par Finally we wish to explore the effects of perturbative as well as\nnon-perturbative contributions on the dissociation of quarkonia states\nqualitatively in terms of the debye mass.\nInstead of lattice Debye mass ($m_D^L$), if we use the leading-order\nDebye mass ($m_D^{\\rm{LO}}<m_D^L$), the screening of the potential \nwill be much smaller and hence the binding energies ($1\/m_{{}_D}^4$) \nwill be enhanced substantially\nand results in the increase of the dissociation temperatures.\nOn the other hand, if we include the non-perturbative corrections \nof order $O(g^2T)$ and $O(g^3T)$ to the leading-order Debye mass~\\cite{kajantie},\nthe dissociation temperatures become unrealistically small\nwhich looks unfeasible. Thus, this study\nprovides us a handle to decipher the extent up to which and how much\nnon-perturbative effects should be incorporated into the Debye mass.\n\n\\section{Conclusions and Outlook}\\label{conclu}\nIn conclusion, we have studied the dissociation of quarkonia \nby correcting the full Cornell potential with a dielectric function \nembodying the effects of an weakly anisotropic medium \nwhere the in-medium modification causes less screening of the interaction\nand hence the potential gets stronger than in an isotropic medium. \nAnisotropy further introduces a characteristic\nangular ($\\theta$) dependence to the potential in the coordinate space, in addition to the \ninter-particle separation $r$, making it spherically non-symmetric \nand needs to be solved numerically by the three-dimensional\nSchr\\\"odinger equation. However, in the small $\\xi$-limit, the spherically \nnon-symmetric component is much smaller in comparison to spherically \nsymmetric component and can be treated in a perturbation theory and\nthe symmetric (isotropic) component is solved numerically by\none-dimensional radial Schr\\\"{o}dinger equation.\nSo the corrected binding energy is obtained by the direction-independent shift \ndue to the spherically non-symmetric component to the eigen values of \nspherically-symmetric potential.\n\nWe have observed that the quarkonia states are always more bound \nand as a consequence, they survive\nhigher temperature compared to the isotropic medium. \nOur results are found relatively higher compared to similar \ncalculation~\\cite{Dumitru08}, which\nmay be due to the absence of three-dimensional medium modification \nof the linear term in their calculation. In fact, one-dimensional Fourier\ntransform of the Cornell potential yields the similar form\nused in the lattice QCD in which one-dimensional color flux tube structure\nwas assumed~\\cite{dixit}.\nHowever, at finite temperature that may not be\nthe case since the flux tube structure may expand in more\ndimensions~\\cite{Satz}.\nTherefore, it would be better to consider the three-dimensional form of the medium\nmodified Cornell potential which has been done exactly in the present work.\n\n\n\nIn brief, $J \/ \\psi$ is found to be dissociated at 1.38 $T_c$ and $1.43~T_c $ \nfor $\\xi$=0 and 0.6, respectively\nwhereas the corresponding temperatures for the $\\Upsilon$ state are\n$1.70~T_c$ and $1.72~T_c$.\nMoreover we explore the effects of perturbative as well as\nnon-perturbative effects on the dissociation process qualitatively.\n{\\em For example}, the perturbative result of Debye mass \ngives much higher values of dissociation temperatures \nwhereas the inclusion of non-perturbative corrections to it\ngives unrealistically smaller values. It may be important to note that in \nthe weakly-coupled regime, the effects of (non-perturbative) terms {\\em viz.}\n$g^2T$, \n$g^3T$ etc. may be checked separately but in the strong-coupling regime, this \nmay not be possible because they are no longer uncoupled.\nThese findings envisage a basic question about the nature of dissociation \nof quarkonium in an anisotropic hot QCD medium.\n\nApart from the uncertainty of the correct form of the potential,\nthere is an arbitrariness in the definition of dissociation temperature. So,\nin future, we wish to investigate the dissociation \nthrough the decay width of quarkonium bound states calculated from the\nimaginary part of the potential because it is now well understood that potential\nin thermal medium has always an imaginary component~\\cite{Laine2,nora11}. \n\n\\noindent {\\bf Acknowledgments:}\nWe are thankful for some financial assistance from CSIR project (CSR-656-PHY),\nGovernment of India. We also thank M G Mustafa for some suggestion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\renewcommand{\\theequation}{1.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\nString and M-theory all suggest that we may live in a world that \nhas more than three spatial dimensions \\cite{string1,string2,string3}. Because only three of these are \npresently observable, one has to explain why the others are hidden from \ndetection.  One such explanation is the so-called Kaluza-Klein (KK) \ncompactification, according to which the\nsize of the extra dimensions is very small (often taken to be on the order\nof the Planck length) \\cite{zcw1,zcw2}.  As a consequence, modes that have momentum in the\ndirections of the extra dimensions are excited at currently inaccessible\nenergies.\n \nIn such a frame of compactifications, one often finds that the low-dimensional \neffective theories are usually singular, and such singularities might disappear\nwhen we consider the problem in the original high dimensional spacetimes.\nIn particular, it was shown that the 4-dimensional extreme black hole with a\nspecial dilaton coupling constant can be interpreted as a completely non-singular,\nnon-dilatonic, black p-brane in (4+p)-dimensions, provided that $p$ is\nodd  \\cite{GHT95}.\n\nRecently, three of the current authors \\cite{TWW08} studied the problem in the frame \nof colliding two timelike branes in string theory. After developing the general \nformulas to describe such events, we studied a particular class of exact solutions \nfirst in the 5-dimensional effective theory, and then lifted it to the 10-dimensional \nspacetime. In general, the 5-dimensional spacetime is singular, due to the mutual \nfocus of the two \ncolliding 3-branes. Non-singular cases also exist, but with the price that both of the\ncolliding branes violate all the three energy conditions, weak, dominant, and strong\n\\cite{HE73}.\nAfter lifted to 10 dimensions,  we found that the spacetime remains singular,\nwhenever it is singular in the 5-dimensional effective theory. In the cases where\nno singularities are formed after the collision, we found that the two 8-branes necessarily\nviolate all the energy conditions. \n\nIn this paper we shall address the same problem, but for the sake of simplicity,\nwe consider only the case where no branes are present.  Specifically, the\npaper is organized as follows. In the next section, we will set up the model to be\nstudied in the framework of both $D+d$-dimensional  string theory and its $D$-dimensional \neffective theory. In Section III, we  study two classes of exact solutions, by paying\nparticular attention on their local and global singular behavior. In Section IV,\nwe first lift these solutions to the $D+d$-dimensional spacetimes of string theory, and \nthen study their singular behavior. Section V contains our conclusions and\ndiscussing remarks. \n\n \n\n\\section{Toroidal Compactification of the Effective Action}\n\n\\renewcommand{\\theequation}{2.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\nLet us consider the toroidal compactification of the NS-NS sector of the action \n in (D+d) dimensions, $\\hat{M}_{D+d} = M_{D}\\times M_{d}$, where for the string \n theory we have\n$D+d = 10$. Then, the action takes the form\n\\cite{LWC001,LWC002,LWC003},\n\\begin{eqnarray}\n\\label{2.1}\nS_{D+d} &=& - \\frac{1}{2\\kappa^{2}_{D+d}}\n\\int{d^{D+d}x\\sqrt{\\left|\\hat{g}_{D+d}\\right|}  e^{-\\hat{\\Gamma}} \\left\\{\n{\\hat{R}}_{D+d}[\\hat{g}]\\right.}\\nonumber\\\\\n& & \\left. + \\hat{g}^{AB}\\left(\\hat{\\nabla}_{A}\\hat{\\Gamma}\\right)\n\\left(\\hat{\\nabla}_{B}\\hat{\\Gamma}\\right) - \\frac{1}{12}{\\hat{H}}^{2}\\right\\},\n\\end{eqnarray}\nwhere in this paper we consider the $(D+d)$-dimensional spacetimes described by the\nmetric,\n\\begin{eqnarray}\n\\label{2.2}\nd{\\hat{s}}^{2}_{D+d} &=& \\hat{g}_{AB} dx^{A}dx^{B} = \n   \\gamma_{\\mu\\nu}\\left(x^{\\lambda}\\right) dx^{\\mu}\ndx^{\\nu} \\nonumber\\\\& & + \\hat{\\Phi}^{2}\\left(x^{\\lambda}\\right) \n\\hat{\\gamma}_{ab}\\left(z^{c}\\right)dz^{a} dz^{b},\n\\end{eqnarray}\nwith  $\\gamma_{\\mu\\nu}\\left(x^{\\lambda}\\right)$ and $\\hat{\\Phi}^{2}\n\\left(x^{\\lambda}\\right)$ depending only on the coordinates  $x^{\\lambda}$ of \nthe spacetime $M_{D}$, and $\\hat{\\gamma}_{ab}\\left(z^{c}\\right)$ only on the internal \ncoordinates $x^{c}$, where $\\mu, \\nu, \\lambda = 0, 1, 2, ...,\nD-1$; $a, b, c = D, D+1, ..., D+d - 1$; and $A, B, C = 0, 1, 2, ...,\nD+ d -1$. Assuming that matter fields are all independent of $z^{a}$, \none finds that the internal space $M_{d}$ must be Ricci flat $R[\\hat{\\gamma}] = 0$ \n\\cite{LWC001,LWC002,LWC003}. For the purpose of the current work, it is sufficient to assume \nthat $M_{d}$ is a $d-$dimensional torus, $T^{d} = S^{1} \\times\nS^{1} \\times ... \\times S^{1}$.  \nThen, we find that \n\\begin{eqnarray}\n\\label{2.3}\n\\hat{R}_{D+d}\\left[\\hat{g}\\right] &=& R_{D}\\left[{\\gamma}\\right] \n+ \\frac{d(d-1)}{\\hat{\\Phi}^{2}} \n\\gamma^{\\mu\\nu}\\left(\\nabla_{\\mu}\\hat{\\Phi}\\right)\\left(\\nabla_{\\nu}\\hat{\\Phi}\\right)\\nonumber\\\\\n& & - \\frac{2}{\\hat{\\Phi}^{d}}\\gamma^{\\mu\\nu} \\left(\\nabla_{\\mu}\\nabla_{\\nu}\\hat{\\Phi}^{d}\\right).\n\\end{eqnarray}\nIgnoring the dilaton  $\\hat{\\Gamma}$ and the form field  $\\hat{H}$, \nthe integration of the action (\\ref{2.1})\nover the internal space yields,\n\\begin{eqnarray}\n\\label{2.4}\nS_{D}^{eff.} &=& - \\frac{1}{2\\kappa^{2}_{D}} \\int{d^{D}x\\sqrt{\\left|\\gamma\\right|}\n \\hat{\\Phi}^{d}\\left\\{R_{D}\\left[\\gamma\\right] \\right.}\\nonumber\\\\\n& & \\left. \n+ \\frac{d(d-1)}{\\hat{\\Phi}^{2}} \n\\gamma^{\\mu\\nu}\\left(\\nabla_{\\mu}\\hat{\\Phi}\\right)\\left(\\nabla_{\\nu}\\hat{\\Phi}\\right)\n\\right\\},\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{2.5}\n\\kappa_{D}^{2} \\equiv \\frac{\\kappa_{D+d}^{2}}{V_{s}},\n\\end{equation}\nand $V_{s}$ is defined as\n\\begin{equation}\n\\label{2.6}\nV_{s} \\equiv \\int{\\sqrt{\\hat{\\gamma}} d^{d}z}.\n\\end{equation}\nFor a string scale compactification, we have $V_{s} = \n\\left(2\\pi \\sqrt{\\alpha'}\\right)^{d}$, where $\\left(2\\pi  \\alpha'\\right)$\nis the inverse string tension.\n\nAfter the conformal transformation,\n\\begin{equation}\n\\label{2.7}\ng_{\\mu\\nu} = \\hat{\\Phi}^{\\frac{2d}{D-2}}\\gamma_{\\mu\\nu},\n\\end{equation}\nthe D-dimensional effective action of Eq.(\\ref{2.4}) can be cast in the\nminmally coupled form,\n\\begin{eqnarray}\n\\label{2.8}\nS_{D}^{eff.} &=&  - \\frac{1}{2\\kappa^{2}_{D}}\\int{d^{D}x\\sqrt{\\left|g_{D}\\right|}\n  \\left\\{ R_{D}\\left[g\\right] \n-  \\kappa^{2}_{D} \\left(\\nabla\\phi\\right)^{2}\\right\\}},\\nonumber\\\\ \n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{2.9}\n\\phi \\equiv \\pm \n  \\left(\\frac{(D+d-2)d}{\\kappa^{2}_{D}\\left(D-2\\right)}\\right)^{\\frac{1}{2}} \n  \\ln\\left(\\hat{\\Phi}\\right).\n\\end{equation}\n\nIn this paper, we refer the actions of Eqs. (\\ref{2.1}) and (\\ref{2.4})  to as  {\\em the string frames},\nand the one of Eq.(\\ref{2.8})   {\\em the Einstein frames}. It should be noted that\nsolutions related by this conformal transformation can have completely different \nphysical and goemetrical properties in the two frames. In particular, in one \nframe a solution can be singular, while in the other it can be totally free from \nany kind of singularities. A simple example is the conformally-flat spacetimes\n$g_{AB} = \\Omega^{2}(x) \\eta_{AB}$, where the spacetime described by $g_{AB}$\ncan have a completely different spacetime structure from that of the Minkowski,\n$\\eta_{AB}$.\n\nAlthough we are mainly interested in the string theory with the split\n$D = 5$ and $d = 5$, in most cases considered in this paper \nwe shall not impose these restrictions here,\nso that our results obtained in this paper can be applied to other situations.\n\n\n\\section{Solutions in D-dimensional Spacetimes in   the   Einstein Frame}\n\n\\renewcommand{\\theequation}{3.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\n \nThe variation of the action (\\ref{2.8}) with respect to $g_{\\mu\\nu}$ and\n$\\phi$ yields the D-dimensional Einstein-scalar field equations,\n\\begin{eqnarray}\n\\label{3.1a}\n& & R_{\\mu\\nu} = \\kappa^{2}_{D} \\phi_{,\\mu} \\phi_{,\\nu},\\\\\n\\label{3.1b}\n& & \\nabla_{\\lambda} \\nabla^{\\lambda} \\phi = 0,\n\\end{eqnarray}\nwhere   $(\\;)_{,\\mu} \\equiv\n\\partial (\\;)\/\\partial x^{\\mu}$.\n\nIn this paper, we consider the D-dimensional spacetimes described by the metric\n\\begin{equation}\n\\label{3.2}\nds^{2}_{D,E} = 2 e^{2\\sigma(u, v)}du dv - e^{2h(u, v)} d\\Sigma^{2}_{D-2},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{3.3}\nd\\Sigma^{2}_{D-2} \\equiv \\sum_{i = 2}^{D-1}{\\left(dx^{i}\\right)^{2}}.\n\\end{equation}\nClearly, the $(D-2)$-dimensional space ${\\cal{S}}$, spaned by  $d\\Sigma^{2}_{D-2}$, \nis Ricci flat, and its topology remains unspecified. In this paper, we assume that \nthis space is compact. One example is that it is a $(D-2)$-dimensional torus.\n\nIt should be noted that metric (\\ref{3.2}) is invariant under the coordinate\ntransformation,\n\\begin{equation}\n\\label{3.8}\nu = f(\\bar{u}), \\;\\;\\;\\; v = g(\\bar{v}),\n\\end{equation}\nwhere $f(\\bar{u})$ and $g(\\bar{v})$ are arbitrary functions of their indicated\narguments. \n\nIntroducing the following two null vectors \\cite{Wang1,Wang2,Wang3,Wang4}, \n\\begin{eqnarray}\n\\label{3.8a}\nl_{\\mu} &\\equiv& \\frac{\\partial u}{\\partial x^{\\mu}} = \\delta^{u}_{\\mu},\\nonumber\\\\\nn_{\\mu} &\\equiv& \\frac{\\partial v}{\\partial x^{\\mu}} = \\delta^{v}_{\\mu}, \n\\end{eqnarray}\nwe can see that both of them are future-directed and orthogonal to ${\\cal{S}}$.\nIn addition, each of these two null vectors defines an affinely parameterized\nnull geodesic congruence,\n\\begin{equation}\n\\label{3.8b}\nl^{\\lambda}\\nabla_{\\lambda}l_{\\mu} = 0 = n^{\\lambda}\\nabla_{\\lambda}n_{\\mu}.\n\\end{equation}\nThen, the expansions of the null ray $u = Const.$ and the one $v = Const.$\nare defined, respectively, by\n\\begin{eqnarray}\n\\label{3.8c}\n\\theta_{l}   &\\equiv& \\nabla^{\\lambda}l_{\\lambda} \n                = \\left(D-2\\right)e^{-2\\sigma}h_{,v},\\nonumber\\\\\n\\theta_{n}   &\\equiv& \\nabla^{\\lambda}n_{\\lambda} \n                = \\left(D-2\\right)e^{-2\\sigma}h_{,u}. \n\\end{eqnarray}\nIt should be noted that the two null vectors are uniquely defined only upto a\nfactor \\cite{HE73,Wang1,Wang2,Wang3,Wang4}.  In fact,  \n\\begin{equation}\n\\label{3.8ca}\n\\bar{l}_{\\mu} = f(u)\\delta^{u}_{\\mu},\\;\\;\\;\n\\bar{n}_{\\mu} = g(v)\\delta^{v}_{\\mu},\n\\end{equation}\nrepresent another set of null vectors that also define affinely parameterized null \ngeodesics, and  the corresponding expansions\nare given by \n\\begin{equation}\n\\label{3.8cb}\n\\bar{\\theta}_{+} = f(u)\\theta_{+},\\;\\;\\;\n\\bar{\\theta}_{-} = g(v)\\theta_{-}. \n\\end{equation}\nHowever, since along each curve $u = Const.\\; \\left(v = Const.\\right)$, the function \n$f(u) \\;\\left(g(v)\\right)$ is constant, this does not affect the definition of \ntrapped surfaces in terms of the expansions (See \\cite{HE73,Wang1,Wang2,Wang3,Wang4} in details). Thus, \nwithout loss of generality, in the following definitions of trapped surfaces and \nblack holes we consider only the expressions given by Eq.(\\ref{3.8c}).\n \n\n{\\em Definitions} \\cite{Pen68,Hay941,Hay942,Wang1,Wang2,Wang3,Wang4}:  The spatial $(D-2)$-dimensional \nsurface ${\\cal{S}}$ of constant $u$ and $v$\nis said {\\em trapped, marginally trapped, or untrapped}, according to whether\n$\\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} > 0$,\n$\\; \\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} = 0$,\nor $\\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} < 0$.\nAssuming that on the marginally trapped surfaces ${\\cal{S}}$ we have\n$\\left.\\theta_{l}\\right|_{{\\cal{S}}} = 0$, then an {\\em apparent horizon} is\nthe closure $\\tilde{\\Sigma}$ of a three-surface $\\Sigma$ foliated by the trapped\nsurfaces $\\cal{S}$ on which $\\left.\\theta_{n}\\right|_{\\Sigma} \\not= 0$.\nIt is said {\\em outer, degenerate, or inner}, according to\nwhether $\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} < 0$,\n$\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} = 0$, or\n$\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} > 0$, where ${\\cal{L}}_{n}$\ndenotes the Lie derivative along the normal direction\n${n}_{\\mu}$, given by,\n\\begin{eqnarray}\n\\label{3.8d}\n{\\cal{L}}_{n}\\theta_{l}   &=& \\left(D-2\\right)e^{-4\\sigma}\n\\left(h_{,uv} - 2\\sigma_{,u}h_{,v}\\right),\\nonumber\\\\\n{\\cal{L}}_{l}\\theta_{n}   &=& \\left(D-2\\right)e^{-4\\sigma}\n\\left(h_{,uv} - 2\\sigma_{,v}h_{,u}\\right). \n\\end{eqnarray}\nIn addition, if $\\left. \\theta_{n}\\right|_{\\Sigma} < 0$\nthen the apparent horizon is said {\\em future}, and if\n$\\left. \\theta_{n}\\right|_{\\Sigma} > 0$ it is said {\\em past}.\n\n{\\em Black holes} are usually defined by the existence of {\\em future outer\napparent horizons} \\cite{HE73,Hay941,Hay942,Ida00,Wang1,Wang2,Wang3,Wang4}. However, in a definition given by\nTipler \\cite{Tip77} the degenerate case was also included.\n\nFor the metric (\\ref{3.2}), we find that Eqs.(\\ref{3.1a}) and (\\ref{3.1b})  yield\n\\begin{eqnarray}\n\\label{3.4a}\n& &  h_{,uu} + {h_{,u}}^{2} - 2 h_{,u}\\sigma_{,u} = \n            - \\frac{\\kappa^{2}_{D}}{D-2}{\\phi_{,u}}^{2},\\\\\n\\label{3.4b}\n& & h_{,vv} + {h_{,v}}^{2} - 2 h_{,v}\\sigma_{,v} = \n           - \\frac{\\kappa^{2}_{D}}{D-2}{\\phi_{,v}}^{2},\\\\\n\\label{3.4c}\n& &  2\\sigma_{,uv} + (D-2)\\left(h_{,uv} + {h_{,u}}h_{,v}\\right) = \n  -  \\kappa^{2}_{D} \\phi_{,u}\\phi_{,v},\\nonumber\\\\\n\\\\\n\\label{3.4d}\n& &   h_{,uv} + (D-2)h_{,u}h_{,v} = 0,\\\\\n\\label{3.4e}\n& &  2\\phi_{,uv} + (D-2)\\left(h_{,u}\\phi_{,v} + h_{,v}\\phi_{,u}\\right) = 0.\n\\end{eqnarray}\nIt should be noted that Eqs.(\\ref{3.4a})-(\\ref{3.4e}) are not all  \nindependent. In fact,\nEq.(\\ref{3.4c}) is the integrability condition of Eqs.(\\ref{3.4a}) and (\\ref{3.4b}), and\ncan be obtained from Eqs.(\\ref{3.4a}), (\\ref{3.4b}) and (\\ref{3.4e}). Therefore,  \nthe field equations reduce to Eqs. (\\ref{3.4a}), (\\ref{3.4b}), (\\ref{3.4d}) and \n(\\ref{3.4e}). To find the solution, one may first integrate Eq.(\\ref{3.4d}) to \nfind $h$, which gives,\n\\begin{equation}\n\\label{3.5}\nh(u, v)  =  \\frac{1}{D-2}\\ln\\left(F(u) + G(v)\\right),\n\\end{equation}\nwhere $F(u)$ and $G(v)$ are arbitrary functions. Then, one can integrate \nEq.(\\ref{3.4e}) to find $\\phi$. Once $h$ and $\\phi$ are found, $\\sigma$ can be \nobtained by integrating  Eqs.(\\ref{3.4a}) and (\\ref{3.4b}). The general solutions\nfor $\\sigma$ and $\\phi$ are unknown. In the following, we shall consider some\nspecific solutions. In particular, we shall consider the three cases separately: \n(a) $F'(u) \\not= 0,\\; G'(v)  = 0$; \n(b) $F'(u) = 0,\\; G'(v)  \\not= 0$; and  (c) $F'(u) G'(v)  \\not= 0$, where a prime\ndenotes the ordinary differentiation.  The second case can be obtained from the first\none by exchanging the $u$ and $v$ coordinates. Thus, without loss of\ngenerality, we need consider only Cases (a) and (c). \n\nBefore proceesing further, we note that for the solution of Eq.(\\ref{3.5}), \n Eqs.(\\ref{3.8c}) and (\\ref{3.8d}) reduce to\n\\begin{eqnarray}\n\\label{3.5a}\n\\theta_{l}   &=&  e^{-2\\sigma}\\frac{G'(v)}{F(u) + G(v)},\\nonumber\\\\\n\\theta_{n}   &=& e^{-2\\sigma}\\frac{F'(u)}{F(u) + G(v)},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\label{3.5b}\n{\\cal{L}}_{n}\\theta_{l}   &=& - \\theta_{l}\\left(\\theta_{n} \n+ 2e^{-2\\sigma}\\sigma_{,u}\\right),\\nonumber\\\\ \n{\\cal{L}}_{l}\\theta_{n}   &=& - \\theta_{n}\\left(\\theta_{l} \n+ 2e^{-2\\sigma}\\sigma_{,v}\\right). \n\\end{eqnarray}\n\n\\subsection{ $F'(u) \\not= 0,\\; G'(v)  = 0$}\n\n\n\nIn this case, from Eq.(\\ref{3.4b}) we find that $\\phi = \\phi(u)$. Hence, Eq.(\\ref{3.4c}) \nyields\n\\begin{equation}\n\\label{3.6}\n\\sigma(u,v) = a(u) + b(u),\n\\end{equation}\nwhere $a(u)$ and $b(u)$ are other arbitrary functions. Using the gauge\nfreedom of Eq.(\\ref{3.8}), without loss of generality we can always set \n$a(u) = 0 = b(u)$, so that this class of solutions are given by\n\\begin{eqnarray}\n\\label{3.7}\n\\sigma(u,v) &=& 0,\\nonumber\\\\\nh(u,v) &=& \\ln\\alpha(u),\\nonumber\\\\\n\\phi(u,v) &=& \\pm \\sqrt{\\frac{D-2}{\\kappa^{2}_{D}}}\n\\int^{u}{\\left(- \\frac{\\alpha''(u')}{\\alpha(u')}\\right)^{1\/2}\ndu'} \\nonumber\\\\\n& & + \\phi_{0},\n\\end{eqnarray}\nwhere $\\alpha(u) \\equiv F(u)^{1\/(D-2)}$, and $\\phi_{0}$ is an integration constant.\nInserting Eq.(\\ref{3.7}) into Eq.(\\ref{3.5a}), we find that\n\\begin{equation}\n\\label{3.7a}\n\\theta_{l}   = 0,\\;\\;\\;  \n\\theta_{n}   = \\left(D-2\\right)\\frac{\\alpha'(u)}{\\alpha(u)},\n\\end{equation}\nfor which we have $\\theta_{l} \\theta_{n} = 0$ identically. Then, according the above\ndefinition, the $(D-2)$-surfaxe ${\\cal{S}}$ is   always marginally trapped. Since\n\\begin{eqnarray}\n\\label{3.7b}\n{\\cal{L}}_{n}\\theta_{l}   =  0 =  {\\cal{L}}_{l}\\theta_{n}, \n\\end{eqnarray}\n${\\cal{S}}$ is also degenerate.  \n\nTo study these solutions further, we notice that \nfor these solutions  all the scalars built from the Riemann \ncurvature tensor are zero, therefore, in the present case scalar curvature singularities\nare always absent \\cite{ES77}.  However, non-scalar curvature singularities might also exist. \nIn particular, tidal forces experienced by an observer may become infinitely large under\ncertain conditions \\cite{HWW02}. To see how this can happen, let us consider the timelike \ngeodesics in the $(u, v)$-plane, which in the present case are simply given by\n\\begin{equation}\n\\label{3.9}\n\\dot{u} = \\gamma_{0},\\;\\;\\;\\;\n\\dot{v} = \\frac{1}{2\\gamma_{0}},\\;\\;\\;\\;\n\\dot{x}^{i} = 0,\n\\end{equation}\nwhere $i = 2, ..., D-1$, $\\; \\gamma_{0}$ is an integration  constant, \nand an overdot denotes the\nordinary derivative with respect to the proper time, $\\lambda$, of the timelike\ngeodesics. Defining $e^{\\mu}_{(0)} = d{x}^{\\mu}\/d\\lambda$, we find that the  \nunit vectors, given by\n\\begin{eqnarray}\n\\label{3.10}\ne^{\\mu}_{(0)} &=& \\gamma_{0}\\delta^{\\mu}_{u} +\n\\frac{1}{2\\gamma_{0}}\\delta^{\\mu}_{v},\\nonumber\\\\\ne^{\\mu}_{(1)} &=& \\gamma_{0}\\delta^{\\mu}_{u} -\n\\frac{1}{2\\gamma_{0}} \\delta^{\\mu}_{v},\\nonumber\\\\\ne^{\\mu}_{(i)} &=&   \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{i}, \n\\end{eqnarray}\nform a freely falling frame, \n\\begin{equation}\n\\label{3.11}\ne^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)} g_{\\mu\\nu} = \\eta_{\\alpha\\beta},\\;\\;\\;\ne^{\\mu}_{(\\alpha); \\nu}e^{\\nu}_{(0)}  = 0,\n\\end{equation}\nwhere $\\eta_{\\alpha\\beta} = {\\mbox{diag.}}\\;\\{- 1, \\;  1, ..., \\;  1\\}$.\nProjecting the Ricci tensor onto the above frame, we find that\n\\begin{eqnarray}\n\\label{3.12}\nR_{(\\alpha)(\\beta)} &\\equiv& R_{\\mu\\nu}e^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)} \\nonumber\\\\\n&=& - \\gamma^{2}_{0}(D-2)  \\left(\\frac{\\alpha''(u)}{\\alpha(u)}\\right) \n\\left(\\delta^{u}_{\\alpha}\\delta^{u}_{\\beta} \\right.\\nonumber\\\\\n& & \\left. -\n\\left(\\delta^{u}_{\\alpha}\\delta^{v}_{\\beta} \n+ \\delta^{v}_{\\alpha}\\delta^{u}_{\\beta}\\right)\n+ \\delta^{v}_{\\alpha}\\delta^{v}_{\\beta}\\right).\n\\end{eqnarray}\nClearly,  the tidal forces remain finite over the whole spacetime, as long as \n$\\alpha''\/\\alpha$ is finite. To see this clearly, let us consider the following\nsolutions,\n\\begin{equation}\n\\label{3.13} \n \\frac{\\alpha''(u)}{\\alpha(u)}  = \n- \\frac{\\omega^{2}}{(u - u_{0})^{\\gamma}},\n\\end{equation}\nfor which we have\n\\begin{eqnarray}\n\\label{3.15}\n\\phi(u,v) &=& \\phi_{0}\n   + \\left(\\frac{\\omega^{2}(D-2)}{\\kappa^{2}_{D}}\\right)^{1\/2}\n\\nonumber\\\\\n& & \\times \\cases{\\frac{2}{2-\\gamma}(u - u_{0})^{1 - \\gamma\/2}, & $ \\gamma \\not= 2$,\\cr\n\\ln{(u - u_{0})}, &  $ \\gamma = 2$,\\cr}\n\\end{eqnarray}\n where $u_{0}$ is an arbitrary constant, and without\nloss of generality, we can always set $u_{0} = 0$, an assumption we shall adopt in the \nfollowing discussions. The constants $\\omega$ and $\\gamma$ have to satisfy the  conditions\n$\\alpha(u) > 0$ and $\\alpha''(u)\/\\alpha(u) < 0$, so that  the metric has the correct \nsigns and the scalar field is real. \n\nFrom Eq.(\\ref{3.9}) we find that $ u   \\sim \n\\gamma_{0}\\lambda$, where the proper time $\\lambda$ was chosen such\nthat $u = 0$ corresponds to $\\lambda = 0$. Then, the distortion,\nwhich is proportional to the twice integrals of $R_{(\\alpha)(\\beta)}$ \nwith respect to the proper time $\\lambda$, is given by\n\\begin{eqnarray}\n\\label{3.14}\nD_{(\\alpha)(\\beta)} &\\equiv& \\int{d\\lambda\\int{R_{(\\alpha)(\\beta)} d\\lambda}} \\nonumber\\\\\n &\\sim& \\cases{ \\lambda\\left(\\ln\\lambda -1\\right), &\n$\\gamma = 1$,\\cr\n\\ln\\lambda, & $\\gamma = 2$,\\cr\n\\lambda^{2-\\gamma}, & $\\gamma \\not= 1, 2$,\\cr}\n\\end{eqnarray}\nas $\\lambda \\rightarrow 0$. To study these solutions further, let us consider\nthe following cases separately.\n\n\\subsubsection{$\\gamma < 0$}\n\nIn this case,  Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) show that both the tidal forces\nand distortions are finite at $u = 0$. Therfore, to have a geodesically maximal spacetime,\nwe need to extend the solutions across this surface to the region $u < 0$. When $\\gamma \n= - n$, where $n (= 1, 2, ...)$ is an integer, the extension is simple, and can be obtained\nby simply taking $u$ to be $u \\in (-\\infty, \\infty)$. However, When $- \\gamma$ is not an\ninteger, the solution is not anyalytical at $u = 0$, and the extension is not unique.\nOne possible extension is to replace $u$ by $|u|$.  Once such an extension is done, it can be\nseen that  both the tidal forces and distortions diverge at $|u| \\rightarrow \\infty$. Therefore,\nthe spacetimes are singular at the null infinities, and the nature of the singularities is\n{\\em strong}, as both of them diverge. The corresponding Penrose diagram is given by Fig. \\ref{fig1a}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1a}\n\\caption{The Penrose diagram for the case $\\gamma < 0$ of the solution (\\ref{3.13})\nin the Einstein frame. The double solid lines $AB$ and $CD$ represent null infinities\n$u = \\pm \\infty$ and denote strong spacetime singularities, where both the tidal \nforces and distortions exerting on a freely falling observer become unbound. The spacetime\nis regular at $u = 0$, and its extension to Region $II$ where $u < 0$ is unique only \nwhen $-\\gamma$ is an integer. \n}\n\\label{fig1a}\n\\end{figure} \n\n\n\n\\subsubsection{$\\gamma = 0$}\n\nIn this case,   $\\alpha(u)$ and $\\phi(u)$ can be given explicitly,  \n\\begin{eqnarray}\n\\label{3.12b}\n \\alpha(u) &=& \\alpha_{0} \\sin\\left(\\omega u + \\Delta\\right),  \\nonumber\\\\\n \\phi(u) &=& \\pm \\sqrt{\\frac{D-2}{\\kappa^{2}_{D}}} \\; \\omega u + \\phi_{0},\n\\end{eqnarray}\nwhere $\\Delta$ and $\\phi_{0}$ are the integration constants. From the\nabove we can see that the $(D-2)$-dimensional space ${\\cal{S}}$ collapses to\na point at $\\omega u + \\Delta = n \\pi$. This can also be seen from the \nexpansion given by  Eq.(\\ref{3.7a}), which  now reads,    \n\\begin{equation}\n\\label{3.12c}\n \\theta_{n} = \\omega \\left(D-2\\right) \n \\frac{\\cos\\left(\\omega u + \\Delta\\right)}{\\sin\\left(\\omega u + \\Delta\\right)}.\n\\end{equation}\nClearly, $\\theta_{n}$ diverges at $\\omega u + \\Delta = n \\pi$. However, a closer\ninvestigation of this solution shows that the spacetime is not singular\nat these points. For example, one may consider the tidal forces measured \nby observers that move along timelike geodesics not perpendicular to ${\\cal{S}}$.\nWithout loss of generality, let us consider the timelike geodesics, described\nby $u = u(\\lambda), \\; v = v(\\lambda), \\; x^{2} = x^{2}(\\lambda)$ and\n$  x^{i} = x^{i}_{0} = Const.$ Then, it can be shown that the timelike geodesical\nequation allows the first integration, and yields,\n\\begin{eqnarray}\n\\label{3.12d}\n \\dot{u} &=& \\gamma_{0}, \\;\\;\\;\n \\dot{v} = \\frac{1}{2\\gamma_{0}}\n           \\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} + 1\\right), \\nonumber\\\\\n\\dot{x}^{2} &=& \\frac{\\beta_{0}}{\\alpha^{2}(u)},\\;\\;\\;\n\\dot{x}^{i} = 0, \\; (i = 3, 4, ..., D-1),  \n\\end{eqnarray}\nwhere $\\gamma_{0}$ and $\\beta_{0}$ are two integration constants.\nFrom the above we find the following unit vectors,\n\\begin{eqnarray}\n\\label{3.10a}\ne^{\\mu}_{(0)} &\\equiv& \\frac{dx^{\\mu}}{d\\lambda\n=\\gamma_{0}\\delta^{\\mu}_{u} + \\frac{1}{2\\gamma_{0}}\n\\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} + 1\\right)\\delta^{\\mu}_{v}\n+ \\frac{\\beta_{0}}{\\alpha^{2}(u)}\\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(1)} &=& \\gamma_{0}\\delta^{\\mu}_{u} + \\frac{1}{2\\gamma_{0}}\n\\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} - 1\\right)\\delta^{\\mu}_{v}\n+ \\frac{\\beta_{0}}{\\alpha^{2}(u)}\\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(2)} &=& \\frac{\\beta_{0}}{\\gamma_{0}\\alpha(u)} \\delta^{\\mu}_{v} \n     + \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(i)} &=&   \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{i}, \n\\; (i = 3, 4, ..., D-1).\n\\end{eqnarray}\nIt can be shown that they satisfy Eq.(\\ref{3.11}), that is, they  also\nform a freely-falling frame. Then, we find that\n\\begin{equation}\n\\label{3.10b}\nR_{(\\alpha)(\\beta)} \\equiv R_{\\mu\\nu}e^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)}  \n= \\omega^{2} \\left(D-2\\right)e^{u}_{(\\alpha)} e^{u}_{(\\beta)},\n\\end{equation}\nwhich is always finite, as can be seen from Eq.(\\ref{3.10a}). Therefore, although\nthe $(D-2)$-dimensional space ${\\cal{S}}$ focuses at $\\alpha(u) = 0$,  \nno spacetime singularities appear at these points, because no trapped compact\nsurface exists in the present case. The Hawking-Penrose singularity theorems\nrequire the existence of both a compact trapped surface and the focusing of\ngeodesics \\cite{HE73}. The extension across these focusing points can be done\nby simply taking $u$ as any real value, i.e.,  $u \\in (-\\infty, \\infty)$.\nOnce such an extension is made, Eq.(\\ref{3.12b}) shows that we may have\nspacetime singularities at $u = \\pm \\infty$.  As Eq.(\\ref{3.14}) shows that\nindeed the distortation at these null infinities diverge, although the \ntidal forces still remain finite. Then, the corresponding Penrsoe diagram\nis given by Fig. \\ref{fig1b}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1b}\n\\caption{The Penrose diagram for the case $\\gamma = 0$ of the solution (\\ref{3.12b})\nin the Einstein frame. The lines $AB$ and $CD$ represent null infinities\n$u = \\pm \\infty$ and the spacetime is singular there, in the sense that\nthe  distortation becomes unbounded along these null infinities, although \nthe tidal forces remain finite.  \n}\n\\label{fig1b}\n\\end{figure} \n\n \n\n\n\n\\subsubsection{$0 < \\gamma < 2$}\n\nIn this case, Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) show that  at\n$u = 0$ the tidal forces become unbounded, while the distortions remain finite.\nThis type of   singularities  is usually said   {\\em weak}, and the spacetime \nbeyond this surface may be extendible \\cite{Ori921,Ori922}, although it is still unclear\nhow to carry out specifically such extensions. As $u \\rightarrow \\infty$,\nfrom Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) we find that the tidal \nforces are bounded, but now the distortions become unbounded. The corresponding\nPenrose diagram is givne by Fig. \\ref{fig1c}.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1c}\n\\caption{The Penrose diagram for the case $0 < \\gamma < 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that the tidal forces become\nunbounded, while the distorations remain bounded. \nIt is also singular at the null infinity $u = \\infty$,  denoted by the line\n$CD$, where the tidal forces remain finite, but the distorations become\nunbounded.  \n}\n\\label{fig1c}\n\\end{figure} \n\n\n\\subsubsection{$ \\gamma = 2$}\n \nWhen $\\gamma = 2$,   Eq.(\\ref{3.13})   has the solution\n\\begin{equation}\n\\label{3.16}\n\\alpha(u) =  \\alpha_{0} u^{\\delta},  \n\\end{equation}\nwhere  $\\omega^{2} = \\delta(1-\\delta)$ with $0 < \\delta < 1$. For such a\nsolution, Eq.(\\ref{3.7a}) reads,    \n\\begin{equation}\n\\label{3.16a}\n \\theta_{n} = \\frac{\\delta \\left(D-2\\right)}{u}.\n\\end{equation}\nThen, from Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) we find that in \nthe present case both  the tidal forces and distortions   become unbound\nat $u = 0$, so a strong spacetime singularity appears at $u = 0$. In this\ncase, we also have $\\theta_{n}(u = 0) = \\infty$. However, at $u = \\infty$,\nthe tidal forces remain finite, while the distortions   become unbound.\nThe corresponding Penrose diagram in this case is given by Fig. \\ref{fig1d}.  \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1d}\n\\caption{The Penrose diagram for the case $  \\gamma = 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that the tidal forces become\nunbounded, while the distorations remain bounded. \nIt is also singular at the null infinity $u = \\infty$,  denoted by the line\n$CD$, where the tidal forces remain finite, but the distorations become\nunbounded.  \n}\n\\label{fig1d}\n\\end{figure} \n\n\\subsubsection{$\\gamma > 2$}\n \n\nWhen $\\gamma >  2$, both the  tidal forces and the distortion\nbecome unbound at $u = 0$, so the spacetime has a strong singularity along \nthis surface. However, at $u = \\infty$ all of them remain finite. Therefore,\nin the present case the spacetime is free of spacetime singularity at the \nnull infinity $ u = \\infty$. The corresponding Penrose diagram in this case \nis given by  Fig. \\ref{fig1e}.\n\n \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1e}\n\\caption{The Penrose diagram for the case $  \\gamma > 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that both the tidal forces\nand distorations become unbounded. At the null infinity $u = \\infty$,  denoted \nby the line $CD$, it is non-singular, because now  both the tidal forces and\ndistorations  remain finite.  \n}\n\\label{fig1e}\n\\end{figure} \n\n\n\n\n\\subsection{ $F'(u) G'(v)  \\not= 0$}\n\nIn this case to solve Eqs.(\\ref{3.4a})-(\\ref{3.4e}), it is found convenient first to\nintruduce two new coordinates $\\bar{u}$ and $\\bar{v}$ via the relations \n$\\bar{u} \\equiv F(u)$ and $\\bar{v} \\equiv G(v)$,  using the gauge freedom (\\ref{3.8}).\nIn terms of these new coordinates, the metric (\\ref{3.2}) takes the form,\n\\begin{eqnarray}\n\\label{3.2a}\nds^{2}_{D,E} &=& \\bar{g}_{\\mu\\nu}d\\bar{x}^{\\mu}d\\bar{x}^{\\nu} \\nonumber\\\\\n&=& 2 e^{2\\Sigma(\\bar{u}, \\bar{v})}d\\bar{u} d\\bar{v} \n- e^{2H(\\bar{u}, \\bar{v})} d\\Sigma^{2}_{D-2},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.3a} \nH(\\bar{u}, \\bar{v}) &\\equiv&  h(u,v)  \n=  \\frac{1}{D-2}\\ln\\left(\\bar{u} + \\bar{v}\\right),\\nonumber\\\\\n\\Sigma(\\bar{u}, \\bar{v}) &\\equiv&     \\sigma({u}, {v}) - \\frac{1}{2}\\ln\\left[F'(u)G'(v)\\right]. \n\\end{eqnarray}\nThen, it can be shown that Eqs.(\\ref{3.4a})-(\\ref{3.4e}) reduce to\n\\begin{eqnarray}\n\\label{3.16aa}\nM_{,t} &=& \\frac{1}{2} t\\left({\\phi_{,t}}^{2} + {\\phi_{,y}}^{2}\\right),\\\\\n\\label{3.16ab}\nM_{,y} &=&   t{\\phi_{,t}}{\\phi_{,y}},\\\\\n\\label{3.16ac}\nM_{,tt} &-& M_{,yy}  = - \\frac{1}{2} \\left({\\phi_{,t}}^{2} - {\\phi_{,y}}^{2}\\right),\\\\\n\\label{3.16ad}\n\\phi_{,tt} &+& \\frac{1}{t} \\phi_{,t} - \\phi_{,yy} = 0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.16ae}\n\\Sigma  &\\equiv& - \\frac{D-3}{2(D-2)} \\ln(t) + \\kappa^{2}_{D} M,\\nonumber\\\\\nt &\\equiv& \\bar{u} + \\bar{v},\\;\\;\\; y \\equiv \\bar{u} - \\bar{v}.\n\\end{eqnarray}\nEq.(\\ref{3.16ac}) is the integrability condition of Eqs.(\\ref{3.16aa}) and (\\ref{3.16ab}).\nThus, once a solution for $\\phi$ is found from Eq.(\\ref{3.16ad}), the remaining is to find $M$ \nfrom Eqs.(\\ref{3.16aa}) and (\\ref{3.16ab}) by quadratures. \n\nUsing the freedom on the choice of the two null vectors defined by Eq.(\\ref{3.8c}), for the\nmetric (\\ref{3.2a}) we define $\\bar{l}_{\\mu} = \\delta^{\\bar{u}}_{\\mu}$ and\n$ \\bar{n}_{\\mu} = \\delta^{\\bar{v}}_{\\mu}$. Then, we find that,\n\\begin{eqnarray}\n\\label{3.16af}\n\\bar{\\theta}_{l} &\\equiv& \\bar{g}^{\\alpha\\beta}\\bar{l}_{\\alpha;\\beta}\n= \\frac{e^{-2\\Sigma}}{t},\\nonumber\\\\\n\\bar{\\theta}_{n} &\\equiv& \\bar{g}^{\\alpha\\beta}\\bar{n}_{\\alpha;\\beta}\n= \\frac{e^{-2\\Sigma}}{t},\n\\end{eqnarray}\nfrom which we find that $\\bar{\\theta}_{l}\\bar{\\theta}_{n} \\ge 0$, where equality holds\nonly when $t = \\infty$ or\/and $\\Sigma = \\infty$. Therefore, in the present case, the\n$(D-2)$-dimensional surface ${\\cal{S}}$ is always trapped or marginally trapped. However,\nit must be noted that the coordinates $(t, y)$ or $(\\bar{u}, \\bar{v})$ do not always\ncover the whole spacetime. There exist cases where the hypersurface $\\Sigma = \\infty$\ndoes not represents the boundary of the spacetime. To have a geodesically maximal \nspacetime, extension beyond this surface is needed. In the extended region(s), one\nmay have $\\bar{\\theta}_{l}\\bar{\\theta}_{n} < 0$, that is, the spacetime is not trapped.\nTypical examples of this kind can be found in \\cite{Wang1,Wang2,Wang3,Wang4}. In the following, we consider \nthree classes of solutions, to be referred, respectively, to as, Class IIa, IIb, \nand IIc, and show explicitly that such cases happen here, too.  \n\n\\subsubsection{Class IIa Solutions}\n\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16a1}\nM    &=& \\frac{1}{2} c^{2} \\ln(t) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln(t) + \\phi_{0},\n\\end{eqnarray}\nwhere $c,\\; \\phi_{0}$ and $M_{0}$ are integration constants. Then, from Eq.(\\ref{3.16af})\nwe find that\n\\begin{equation}\n\\label{3.16a2}\n\\bar{\\theta}_{l} = \\bar{\\theta}_{n} = \\frac{e^{-\\kappa^{2}_{D}M_{0}}}{t^{\\chi^{2} +\n\\frac{1}{D-2}}},\n\\end{equation}\nwhich means that $t = 0$ represents a focusing point, where \n$\\chi \\equiv c^{2}\\kappa_{D}^{2}$. Since now all the surfaces\n${\\cal{S}}$ of constant $t$ and $y$ are trapped, then, according to the singularity\ntheorems \\cite{HE73}, the spacetime must be singular at $t = 0$, as can be seen clearly\nfrom the expression,\n\\begin{equation}\n\\label{3.16a3}\nR_{D}[g] \\equiv   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}= \\frac{A_{0}}{t^{\\chi^{2} + \\frac{D-1}{D-2}}},\n\\end{equation}\nwhere $A_{0} \\equiv 2\\chi^{2} e^{-2\\kappa^{2}_{D}M_{0}}$.\n The corresponding Penrose diagram is given by \nFig. \\ref{fig2a}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2a}\n\\caption{The Penrose diagram for the solutions given by Eq.(\\ref{3.16a1}) in D-dimensional\nspacetime. The horizontal line $t = 0$ represents a big bang  singularity.\nThe $(D-2)$-dimensional surfaces of constant $\\bar{u}$\nand $\\bar{v}$ are always trapped.}\n\\label{fig2a}\n\\end{figure} \n\n\n\\subsubsection{Class IIb Solutions}\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16a4}\nM &=& \\frac{1}{2} c^{2} \\ln\\left(\\frac{t^{4}}{\\left(y^{2} - t^{2}\\right)\n\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2}}\\right) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln\\left(\\frac{t^{2}}{y + \\sqrt{y^{2} - t^{2}}}\\right) + \\phi_{0},\n\\end{eqnarray}\nfor which from Eq.(\\ref{3.16a2}) we find that\n\\begin{equation}\n\\label{3.16ah}\n\\bar{\\theta}_{l} = \\bar{\\theta}_{n} = \n\\frac{\\left\\{\\left(y^{2} - t^{2}\\right)\\left[y\n+ \\sqrt{y^{2} - t^{2}}\\right]^{2}\\right\\}^{\\chi^{2}}}\n{e^{2\\kappa^{2}_{D}M_{0}}t^{4\\chi^{2} +\n\\frac{1}{D-2}}}.\n\\end{equation}\nClearly, in the present case the hypersurface $t = 0$ still represents a focusing\npoint. In addition, on the hypersurfaces $y^{2} = t^{2}$, we have $\\Sigma = \\infty$\nand $\\bar{\\theta}_{l} \\bar{\\theta}_{n} = 0$. Thus, the surfaces ${\\cal{S}}$ now become\nmarginally trapped along $y^{2} = t^{2}$. If the spacetime is not singular on \nthese null surfaces, extension beyond them is needed. To study the singular behavior \nalong these surfaces, let us consider the quantity,\n\\begin{eqnarray}\n\\label{3.16ai}\nR_{D}[g] &\\equiv&   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}\\nonumber\\\\\n&=&  \\frac{2A_{0}\\left(4y^{2} - 3 t^{2} + 2\\right)}\n{\\left[\\sqrt{y^{2} - t^{2}}\\left(y + \\sqrt{y^{2} - t^{2}}\\right)\\right]^{1-2\\chi^{2}}}\\nonumber\\\\\n& & \\times t^{- \\left(4\\chi^{2} + \\frac{D-1}{D-2}\\right)}.\n\\end{eqnarray}\nClearly, the spacetime is singular at $t = 0$. The singular behavior along the hypersurfaces\n$y^{2} = t^{2}$ depend on the values of $\\chi$.\n\n\n\\noindent{\\bf Case B.2.1: $\\chi^{2} < \\frac{1}{2}$}. In this case, \n it is also singular on the null hypersurfaces $y^{2} =  t^{2}$, and the \ncorresponding Penrose diagram is given by Fig. \\ref{fig2b}. \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2b}\n\\caption{The Penrose diagram in the D-dimensional spacetime \nfor the solutions given by Eq.(\\ref{3.16a4}),\nas well as for the ones given by Eq.(\\ref{3.16am}),\nfor $\\chi^{2} < 1\/2$. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$.\nThe spacetime along the line $0D$ and $0E$ are singular.   The two regions\n$I$ and $I'$ are physically disconnected in both cases. \n}\n\\label{fig2b}\n\\end{figure} \n\n\\noindent{\\bf Case B.2.2: $\\frac{1}{2} \\le \\chi^{2} < 1$.} In this case,  \nthe spacetime is not singular at  $y = \\pm t$, but the metric\ncoefficient $\\Sigma$ is. To extend the metric beyond these surfaces, \nbecause of the symmetry, it is suffcient to consider   the extension \nonly across the hypersurface $y = t$ from the region $\\bar{u} > 0,\\; \n\\bar{v} < 0$. The extension across the hypersurface $y = -t$ from the region \n$\\bar{u} < 0,\\; \\bar{v} > 0$ is similar, and can be obtained by exchanging the \nroles of $\\bar{u}$ and $\\bar{v}$. Then, using the gauge freedom \n(\\ref{3.8}), we  introduce two new coordinates \n$\\tilde{u}$ and $\\tilde{v}$  via the relations,\n\\begin{equation}\n\\label{3.16aj}\n \\bar{u} =  \\tilde{u}^{2n},  \\;\\;\\;\n \\bar{v} =  - \\left(-\\tilde{v}\\right)^{2n}, \n \\end{equation}\nwhere \n\\begin{equation}\n\\label{3.16ak}\nn \\equiv \\frac{1}{2(1- \\chi^{2})}.\n\\end{equation}\nIt can be shown that the solutions are analytical across the hypersurfaces $y = \\pm t$ \nonly when $n$ is an integer. Otherwise, the solutions are not analytical, and the extension\nacross $y = \\pm t$ in not unique. Therefore, in the following we shall consider only\nthe case where $n$ is an integer. Then, in terms of $\\tilde{u}$ and $\\tilde{v}$, the\nsolutions read\n\\begin{eqnarray}\n\\label{3.2aa}\nds^{2}_{D,E} &=& \\tilde{g}_{\\mu\\nu}d\\tilde{x}^{\\mu}d\\tilde{x}^{\\nu} \\nonumber\\\\\n&=& 2 e^{2\\tilde{\\Sigma}} d\\tilde{u} d\\tilde{v} \n- e^{2\\tilde{H}} d\\Sigma^{2}_{D-2},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.3a1} \n\\tilde{H} &=&    \\frac{1}{D-2}\\ln\\left(\\tilde{u}^{2n} - \\left(-\\tilde{v}\\right)^{2n}\\right),\\nonumber\\\\\n\\tilde{\\Sigma} &=&    \\left(2\\chi^{2} - \\frac{D-3}{2(D-2)}\\right)\n                     \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\\nonumber\\\\\n\t& & - \\frac{D-3}{2(D-2)} \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\n\t    + \\tilde{\\Sigma}_{0},\\nonumber\\\\\n\\phi &=&  \\frac{2\\chi}{\\kappa_{D}} \n                     \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\n\t\t     + \\phi_{0}, \n\\end{eqnarray}\nwhere $\\tilde{\\Sigma}_{0} \\equiv \\kappa^{2}_{D}M_{0} + \\ln\\left(2^{3\/2 - \\chi^{2}} n\\right)$.\nClearly, the coordinate singularity at $y = t$ or $\\tilde{v} = 0$ disappears, and the solutions \ncan be considered as   valid for $\\tilde{v} > 0$.  To study the properties of the spacetime\nin the extended region, let us consider the quantities, \n\\begin{eqnarray}\n\\label{3.3aa} \n\\tilde{\\theta}_{l} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{v}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\left(-\\tilde{v}\\right)^{2n-1}}{\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}},\\nonumber\\\\\n\\tilde{\\theta}_{n} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{u}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\tilde{u}^{2n-1}}{\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}},\\nonumber\\\\\n{\\cal{L}}_{n}\\tilde{\\theta}_{l}\t &=& - 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\tilde{u}^{n-1}\\left(-\\tilde{v}\\right)^{2n-1}}{\\left(\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}\\right)^{2}}\\nonumber\\\\\n& & \\times\\left\\{\\left(2\\chi^{2} + \\frac{1}{D-2}\\right)\\tilde{u}^{n} \n    + 2\\chi^{2}\\left(-\\tilde{v}\\right)^{n}\\right\\},\\nonumber\\\\    \nR_{D}\\left[\\tilde{g}\\right] &=& 8 n^{2}\\chi^{2}e^{-2\\tilde{\\Sigma}_{0}}\\nonumber\\\\\n& & \\times \\frac{\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)^{\\frac{D-3}{D-2}}\n\\left(-\\tilde{u}\\tilde{v}\\right)^{n-1}}\n{\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)^{4\\chi^{2} + \\frac{D-1}{D-2}}}.\n\\end{eqnarray}\nThen, we find that\n\\begin{eqnarray}\n\\label{3.3ab}\n\\tilde{\\theta}_{l} \\tilde{\\theta}_{n} &=& 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\left(-\\tilde{u} \\tilde{v}\\right)^{2n-1}}\n{\\left(\\tilde{u}^{2n} - \\left(-\\tilde{v}\\right)^{2n}\\right)^{2}}\\nonumber\\\\\n&=&\\cases{< 0, \\tilde{u} \\tilde{v} >0,\\cr\n= 0, \\tilde{u} \\tilde{v} = 0,\\cr\n> 0, \\tilde{u} \\tilde{v} < 0.\\cr}\n\\end{eqnarray}\nThat is, now in the extended region where $\\tilde{u} \\tilde{v} >0$ the  spacetime\nbecomes untrapped. Across the hypersurface $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$,\nwe have\n\\begin{eqnarray}\n\\label{3.3ac}\n& & \\tilde{\\theta}_{l}\\left(\\tilde{u}>0, \\tilde{v} = 0\\right) = 0,\\;\\;\\;\n\\tilde{\\theta}_{n}\\left(\\tilde{u}>0, \\tilde{v} =0 \\right)   >0,\\nonumber\\\\\n& & {\\cal{L}}_{n}\\tilde{\\theta}_{l} \\left(\\tilde{u}> 0, \\tilde{v} = 0\\right) = 0.\n\\end{eqnarray}\nTherefore, the half infinite line $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$ represents\na past degenerate apparent horizon. \n\nTo study the singular behavior of the spacetime in the extended region,\nwe need to distinguish the case where $n$ is an even interger from the one where $n$\nis an odd integer.  When $n$ is an even interger, Eq.(\\ref{3.3aa}) shows that the spacetime\nis singular along the line $\\tilde{u} = \\tilde{v}$, denoted by the vertical line\n$0C$ in Fig. \\ref{fig2c}. \n\nBecause of the symmetry of the spacetime, one can make\na similar extension across the half line $\\bar{u} = 0$ and $\\bar{v} \\ge 0$, but\nnow with\n\\begin{equation}\n\\label{3.16ak2}\n \\bar{u} =  - \\left(-\\tilde{u}\\right)^{2n},  \\;\\;\\;\n \\bar{v} =   \\tilde{v}^{2n}.\n \\end{equation}\nThen, one finds that  this half line also  represents a past degenerate apparent\nhorizon, and the extended region $II'$ is not trapped, although it is disconnected\nwith Region $II$, because of the timelike singularity along the vertical line\n$0C$.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2c}\n\\caption{The Penrose diagram in the D-dimensional spacetime for $\\frac{1}{2} \\le \n\\chi^{2} < 1$ for the solutions given by Eq.(\\ref{3.2aa}) with\n$n$ being an even integer, as well as for the ones  given by Eq.(\\ref{3.16am}) with\n $n$ being an odd integer. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$,\nbut not in Regions $II$ and $II'$. The spacetime along the vertical\nline $0C$ is singular.   The two regions\n$II$ and $II'$ are physically disconnected. The lines $0E$ and $0D$ represent\n past degenerate apparent horizons. \n}\n\\label{fig2c}\n\\end{figure} \n\nWhen $n$ is an odd interger, Eq.(\\ref{3.3aa}) shows that the spacetime\nis not singular along the line $\\tilde{u} = \\tilde{v}$, as shown in Fig. \\ref{fig2d},\nwhere the lines $0E$ and $0D$ represent past degenerate apparent horizons.\nThen, Regions $I$ and $I'$ act as white holes. Since the solutions are symmetric\nwith respect to $t$, one may consider the case where $t \\le 0$. Then, the spacetime\nwill be given by the lower half part of Fig. \\ref{fig2d}, in which the $(D-2)$-dimensional\nsurfaces ${\\cal{S}}$ of constant $\\tilde{u}$ and $\\tilde{v}$ are trapped in\nRegions $III$ and $III'$, and is not trapped in   Region $IV$. The lines $0D'$\nand $0E'$ act as future degenerate apparent horizons, so  Regions $III$ and $III'$ \nnow represent black holes. the spacetimes of the lower half part is disconnected with \nthat of the upper half by the spacelike singularity located along $t = 0$.  \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2d}\n\\caption{The Penrose diagram in the D-dimensional spacetime for $\\frac{1}{2} \\le \n\\chi^{2} < 1$ for the solutions given by Eq.(\\ref{3.2aa}) with\n$n$ being an odd integer, as well as for the ones  given by Eq.(\\ref{3.16am}) with\n $n$ being an even integer. The horizontal line $t = 0$ \nrepresents a spacetime  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\tilde{u}$ and $\\tilde{v}$ are always trapped in Regions $I$,\n$I'$, $III$ and $III'$, but not in Regions $II$, and $IV$.   The upper\nhalf part is disconnected with the lower half part by the spacetime\nsingularity along the line $AB$ where $t=0$. The lines $0E$ and $0D$ represent\n past degenerate apparent horizons, while the ones   $0E'$ and $0D'$ represent\n future degenerate apparent horizons. Regions $I$ and $I'$ represent white holes,\n while Regions $III$ and $III'$ represent black holes.\n}\n\\label{fig2d}\n\\end{figure} \n\n\n\n\n\\noindent{\\bf Case B.2.3: $\\chi^{2} \\ge 1$.}\nWhen $\\chi^{2} \\ge 1$, the hypersurfaces $y = \\pm t$  represent spacetime null infinities, \nand the solutions are already geodesically maximal. Indeed, it is found that the null goedesics \n$\\bar{u} = $ Constant have the integral,\n\\begin{equation}\n\\label{3.16al}\n\\eta = \\cases{\\eta_{0} \\left(-\\bar{v}\\right)^{1- \\chi^{2}}, & $\\chi^{2} > 1$,\\cr\n\\eta_{0} \\ln(- \\bar{v}), & $\\chi^{2} = 1$,\\cr}\n\\end{equation}\nnear the hypersurface $ y = t\\; (\\bar{v} = 0)$, where $\\eta_{0}$ is an integration constant, and\n$\\eta$ denotes the affine parameter along the null geodesics.\nThus, as $\\bar{v} \\rightarrow 0^{-}$, we always have $|\\eta| \\rightarrow \\infty$. \n\nSimilar, it can be shown that the hypersurface $y = -t$ also represents a null infinity of the \nspacetime. Therefore, the \ncorresponding Penrose diagram  is given by  Fig. \\ref{fig2e}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2e}\n\\caption{The Penrose diagram for the solutions given by Eq.(\\ref{3.16a4}) and\nfor the ones  given by Eq.(\\ref{3.16am})\nin the D-dimensional spacetime for $\\chi^{2} \\ge 1$. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$,\nand are physically disconnected in both cases. The  line $0D$ and $0E$ represent \nspacetime null infinities, and are not singular.   \n}\n\\label{fig2e}\n\\end{figure} \n\n\\subsubsection{Class IIc Solutions}\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16am}\nM &=& \\frac{1}{2} c^{2} \\ln\\left(\\frac{\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2}}\n{y^{2} - t^{2}}\\right) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln\\left(y + \\sqrt{y^{2} - t^{2}}\\right) + \\phi_{0},\n\\end{eqnarray}\nfor which we find\n\\begin{eqnarray}\n\\label{3.16ana}\n\\bar{\\theta}_{l} &=& \\bar{\\theta}_{n} = \\frac{e^{-2\\kappa^{2}_{D}M_{0}}\n         \\left(y^{2} - t^{2}\\right)^{\\chi^{2}}}\n\t {t^{\\frac{1}{D-2}} \\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2\\chi^{2}}}\\nonumber\\\\\n&\\propto& \\cases{ t^{-\\frac{1}{D-2}}, & $y > 0$,\\cr\nt^{-\\left(4\\chi^{2} + \\frac{1}{D-2}\\right)}, & $ y < 0$,\\cr}\n\\end{eqnarray}\nas $ t \\rightarrow 0$. Therefore, in this case the hypersurface $t =0$ represents a \nfocusing point, while the ones $y = \\pm t$ may represent some kind of horizons. \nTo see their exact natures, let us consider the quantity,\n\\begin{eqnarray}\n\\label{3.16an}\nR_{D}[g] &=&   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}   \\nonumber\\\\\n&=& - \\frac{2A_{0}\\left(y^{2} -  t^{2}\\right)^{\\chi^{2} -1\/2}}\n{\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2\\chi^{2}+1}\n\\; t^{\\frac{D-3}{D-2}}.\n\\end{eqnarray}\nThus, the nature of the spacetime singularity at $t = 0$ is different on the half line\n$y > 0$  from that on the other half line  $y < 0$. In particular, we find\n\\begin{eqnarray}\n\\label{3.16anb}\nR_{D}[g]  &\\propto& \\cases{ - t^{\\frac{D-3}{D-2}} \\rightarrow 0, & $y > 0$,\\cr\nt^{-\\left(4\\chi^{2} + \\frac{1}{D-2}\\right)} \\rightarrow - \\infty, & $ y < 0$,\\cr} \n\\end{eqnarray}\nas  $ t \\rightarrow 0$. Therefore, $R_{D}[g]$ is singular only on the half\nline $t = 0, \\; y \\le 0$. However, the studies of other scalars show that the\nother half line, $t = 0$ and $y > 0$, is also singular. For example, the \ncorresponding Kretschmann scalar is given by\n\\begin{eqnarray}\n\\label{Kretsch}\nI_{D} &\\equiv& R^{\\mu\\nu\\lambda\\sigma}R_{\\mu\\nu\\lambda\\sigma}\\nonumber\\\\\n&=&\\left. \\frac{\\left(y^{2} - t^{2}\\right)^{2\\chi^{2} -5}}{8 t^{3}\n\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{4\\chi^{2}}}I^{(0)}_{D}(t,y)\\right|_{D=4},\n\\end{eqnarray}\nwhere, as $t \\rightarrow 0$, we have\n\\begin{equation}\n\\label{Kretsch2}\nI^{(0)}_{4}  \\simeq \\cases{- 9y^{10}, & $y > 0$,\\cr\n- \\left(1024\\chi^{6} -320\\chi^{4} + 96 \\chi^{2} + 9\\right)y^{10}, & $y < 0$.\\cr}\n\\end{equation}\n \n \nOn the other hand, depending on the value of $\\chi$, the spacetime may or may not\nbe singular on the  two null surfaces $y = \\pm t$, similar to the last case.\nIn particular,  when $\\chi^{2} < 1\/2$, the spacetime is singular there,   \nand the the corresponding Penrose diagram is also given  by Fig. \\ref{fig2b}.\n\n \n \nWhen $ \\frac{1}{2} \\le \\chi^{2} < 1$,  the spacetime is not singular at  \n$y = \\pm t$, although the metric coefficients are. Then, extending the solutions\nto the region $\\tilde{v} > 0$ is needed. As in the last case, the extension\nacross this surface is uniqe only when $n$ defined by Eq.(\\ref{3.16ak}) is an\ninteger. Thus, in the following we shall consider only this case. Then, the \nextension can  be done by  introducing the new  coordinates  $\\tilde{u}$ and\n$\\tilde{v}$ defined by Eq.(\\ref{3.16aj}), for which we find the extended metric \ntakes the same form as that given by Eq.(\\ref{3.2aa}), but now with \n\\begin{eqnarray}\n\\label{3.3a1a} \n\\tilde{H} &=&    \\frac{1}{D-2}\\ln\\left(\\tilde{u}^{2n} - \\tilde{v}^{2n}\\right),\\nonumber\\\\\n\\tilde{\\Sigma} &=&    \\left(2\\chi^{2} - \\frac{D-3}{2(D-2)}\\right)\n                     \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\\nonumber\\\\\n\t& & - \\frac{D-3}{2(D-2)} \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\n\t    + \\tilde{\\Sigma}_{0},\\nonumber\\\\\n\\tilde{\\Sigma}_{0} &\\equiv& \\kappa^{2}_{D}M_{0} \n  + \\frac{1}{2}\\ln\\left(4^{1 - \\chi^{2}} n^{2}\\right),\\nonumber\\\\\n  \\phi &=&  2c    \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\n\t\t     + \\phi_{0}.\t\t     \n\\end{eqnarray} \nThus, the coordinate singularity at  $\\tilde{v} = 0$ disappears, and the solutions \ncan be considered as   valid for $\\tilde{v} > 0$, too.  In terms of $\\tilde{u}$ and \n$\\tilde{v}$, we find   \n\\begin{eqnarray}\n\\label{3.3aa2} \n\\tilde{\\theta}_{l} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{v}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\left(-\\tilde{v}\\right)^{2n-1}}{\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}},\\nonumber\\\\\n\\tilde{\\theta}_{n} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{u}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\tilde{u}^{2n-1}}{\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}},\\nonumber\\\\\n{\\cal{L}}_{n}\\tilde{\\theta}_{l}\t &=& - 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\tilde{u}^{n-1}\\left(-\\tilde{v}\\right)^{2n-1}}{\\left(\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}\\right)^{2}}\\nonumber\\\\\n& & \\times\\left\\{\\left(2\\chi^{2} + \\frac{1}{D-2}\\right)\\tilde{u}^{n} \n    - 2\\chi^{2}\\left(-\\tilde{v}\\right)^{n}\\right\\},\\nonumber\\\\    \nR_{D}\\left[\\tilde{g}\\right] &=& - 4^{\\chi^{2} + \\frac{1}{2}}\n\\chi^{2}  e^{-2\\kappa^{2}_{D}M_{0}} \\nonumber\\\\\n& & \\times \\frac{\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)^{\\frac{D-3}{D-2}}\n\\left(-\\tilde{u}\\tilde{v}\\right)^{n-1}}\n{\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)^{4\\chi^{2} + \\frac{D-1}{D-2}}}.\n\\end{eqnarray}\nFrom the above expressions, we can see that  in the extended region where \n$\\tilde{u} \\tilde{v} >0$ the  spacetime\nbecomes untrapped. Across the hypersurface $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$,\nwe have $\\tilde{\\theta}_{l} = 0,\\; \\tilde{\\theta}_{n}   >0$ and \n$ {\\cal{L}}_{n}\\tilde{\\theta}_{l}   = 0$. That is, the half infinite line \n$\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$ acts as a past degenerate apparent horizon. \n\nIn addition, in contrast to the last case, the expression of $R_{D}\\left[\\tilde{g}\\right]$\ngiven above tells us that the spacetime is singular along the vertical line\n$\\tilde{u} = \\tilde{v}$ for $n$ being an odd integer, and not singular for \n$n$ being an even integer. The corresponding Penrose diagram is given, respectively, by\nFigs. \\ref{fig2b} and \\ref{fig2c}. \n\n \n\n\nWhen $\\chi^{2} \\ge 1$, the hypersurfaces $y = \\pm t$ already represent the null \ninfinities  of the spacetime, \nand the corresponding Penrose diagram  is that \nof Fig. \\ref{fig2e}, where the hypersurrfaces $\\bar{u} = 0$ and  \n$\\bar{v} = 0$  are not singular, and\nthe two regions, $II$ and $II'$, are still not connected.\n\n \n\n\n\\section{Solutions  in ($D+d$)-dimensional Spacetimes in the string Frame} \n\n\\renewcommand{\\theequation}{4.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n \nIn ($D+d$)-dimensions, the metric in the string frame takes the form of Eq. (\\ref{2.2}). Considering\nEq. (\\ref{3.2}), we find that it can be written in the form,\n\\begin{eqnarray}\n\\label{8.1}\nd\\hat{s}^{2}_{D+d} &=& \\hat{g}_{AB}dx^{A} dx^{B} = \\gamma_{\\mu\\nu}dx^{\\mu}dx^{\\nu} + \\hat{\\Phi}\\hat{\\gamma}_{ab}dz^{a}dz^{b}\\nonumber\\\\\n&=& 2e^{2\\hat{\\sigma}(\\hat{u}, \\hat{v})}d\\hat{u}d\\hat{v} - e^{2\\hat{h}(\\hat{u}, \\hat{v})}d\\Sigma_{D-2}^{2} \\nonumber\\\\\n& & \n- e^{2\\hat{g}(\\hat{u}, \\hat{v})}\\hat{\\gamma}_{ab}dz^{a}dz^{b},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.17a}\n\\gamma_{\\mu\\nu} &=& \\exp\\left\\{\\epsilon_{a}\n\\left(\\frac{4\\kappa^{2}_{D}d}{(D-2)(D+d-2)}\\right)^{1\/2}\\phi\\right\\}\n g_{\\mu\\nu},\\nonumber\\\\\n\\hat{\\Phi}  &=& e^{\\hat{g}} =  \\exp\\left\\{- \\epsilon\n\\left(\\frac{(D-2)\\kappa^{2}_{D}}{(D+d-2)d}\\right)^{1\/2}\\phi\\right\\},\n\\end{eqnarray}\n as can be seen from Eqs.(\\ref{2.7}) and (\\ref{2.9}), where $\\epsilon_{a} = \\pm 1$.\n \n Introducing the two null vectors, $\\hat{l}_{A}$ and $\\hat{n}_{A}$ via\n \\begin{equation}\n \\label{8.2}\n \\hat{l}_{A} = \\delta^{\\hat{u}}_{A}, \\;\\;\\;  \\hat{n}_{A} = \\delta^{\\hat{v}}_{A},\n \\end{equation}\n once can show that they also define two null affinely defined geodesic congruences,\n $ \\hat{l}_{A; B} \\hat{l}^{B} =  0 =  \\hat{n}_{A; B} \\hat{n}^{B}$. The corresponding expansions\n are given by\n \\begin{eqnarray}\n \\label{8.3}\n \\hat{\\theta}_{l} &\\equiv&  \\hat{l}_{A; B}\\hat{g}^{AB}\n    =e^{-2\\hat{\\sigma}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{v}} + d \\hat{g}_{,\\hat{v}}\\Big],\\nonumber\\\\\n  \\hat{\\theta}_{n} &\\equiv&  \\hat{n}_{A; B}\\hat{g}^{AB}\n   = e^{-2\\hat{\\sigma}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{u}} + d \\hat{g}_{,\\hat{u}}\\Big],\n \\end{eqnarray}\n from which we find\n  \\begin{eqnarray}\n \\label{8.4}\n{\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l} &=&   e^{-4\\hat{\\sigma}}\\Bigg\\{\\big(D-2\\big)\\hat{h}_{,\\hat{u}\\hat{v}} + d \\hat{g}_{,\\hat{u}\\hat{v}}\\nonumber\\\\\n& & ~~~~~~~~\n - 2\\hat{\\sigma}_{,\\hat{u}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{v}} + d \\hat{g}_{,\\hat{v}} \\Big]\\Bigg\\},\\nonumber\\\\\n{\\cal{L}}_{\\hat{l}}  \\hat{\\theta}_{n} &=&   e^{-4\\hat{\\sigma}}\\Bigg\\{\\big(D-2\\big)\\hat{h}_{,\\hat{u}\\hat{v}} + d \\hat{g}_{,\\hat{u}\\hat{v}}\\nonumber\\\\\n& & ~~~~~~~~\n - 2\\hat{\\sigma}_{,\\hat{v}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{u}} + d \\hat{g}_{,\\hat{u}} \\Big]\\Bigg\\}.\n  \\end{eqnarray}\n\n \n\n\n\\subsection{$F'(u) \\not= 0,\\; G'(v)  = 0$}\n\nIn this case, the modulus  $\\phi(u,v)$ is given by Eq.(\\ref{3.15}). \nIn the $(D + d)$-dimensional \nspacetime, the metric can be written in the form\n\\begin{eqnarray}\n\\label{3.30}\nd{\\hat{s}}^{2}_{D+d} &=&  2d\\hat{u}d\\hat{v} \n- e^{2\\hat{h}(\\hat{u})}d^{2}\\Sigma_{D-2}\\nonumber\\\\\n& & \\;\n  - \\hat{\\Phi}^{2}(\\hat{u}) \\hat{\\gamma}_{ab}(z) dz^{a} dz^{b},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.30a}\nd\\hat{u}  &\\equiv& e^{2\\hat{\\sigma}}du,\\nonumber\\\\\n\\hat{\\sigma} &\\equiv& \\sigma - \\frac{d}{D-2}\\ln\\hat{\\Phi},\\nonumber\\\\\n\\hat{h} &\\equiv& h - \\frac{d}{D-2}\\ln\\hat{\\Phi}.\n\\end{eqnarray}\nFollowing what we did in the Einstein frame, we can construct\na free-falling frame in the $(D+d)$-dimensions, given by \n\\begin{eqnarray}\n\\label{3.34}\ne^{A}_{(0)} &=& \\hat{\\gamma}_{0}\\delta^{A}_{\\hat{u}} \n+ \\frac{1}{2\\hat{\\gamma}_{0}}\\delta^{A}_{v},\\nonumber\\\\\ne^{A}_{(1)} &=& \\hat{\\gamma}_{0}\\delta^{A}_{\\hat{u}} \n- \\frac{1}{2\\hat{\\gamma}_{0}}\\delta^{A}_{v},\\nonumber\\\\\ne^{A}_{(i)} &=& e^{-\\hat{h}}  \\delta^{A}_{i},\\nonumber\\\\\ne^{A}_{(b)} &=& \\hat{\\Phi}^{-1}  \\delta^{A}_{b}, \n\\end{eqnarray}\n which satisfies the relations,\n\\begin{equation}\n\\label{3.35}\ne^{A}_{(C)}e^{B}_{(D)} \\hat{g}_{AB} = \\eta_{CD},\\;\\;\\;\ne^{A}_{(C); B} e^{B}_{(0)} = 0,\n\\end{equation}\nwhere $\\hat{\\gamma}_{0}$ is another integration constant.\nThen, it can be shown that the Riemann tensor in this case has only two \nindependent components, given by\n\\begin{eqnarray}\n\\label{3.33}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=& \n- \\hat{\\gamma}_{0}^{2}\\left(\\hat{h}_{,\\hat{u}\\hat{u}}\n+  \\hat{h}_{,\\hat{u}}^{2}   \\right)\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{ \\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2}\\left(\\frac{\\hat{\\Phi}_{,\\hat{u}\\hat{u}}}\n{\\hat{\\Phi}}\\right)\\delta^{a}_{b}. \n\\end{eqnarray}\n\nIn addition, Eqs.(\\ref{8.3}) and (\\ref{8.4}) now read,\n\\begin{eqnarray}\n\\label{3.33aa}\n\\hat{\\theta}_{n} &=& \\big(D-2\\big) \\alpha'(u)\\exp\\Big(-a u^{(2-\\gamma)\/2}\\Big), \\nonumber\\\\\n\\hat{\\theta}_{l} &=& 0, \\;\\;\\; {\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l} = 0 = {\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l}.\n\\end{eqnarray}\nThen, according the previous\ndefinition, the $(D+d-2)$-surfaxe ${\\cal{S}}$ is   always marginally trapped and degenerate.  \n\n\n\nTo study the solutions further, it is found convenient to\nconsider the two cases $\\gamma \\not= 2$ and $\\gamma =2$ separately.\n\n\n\n\n\n\n\n\\subsubsection{$\\gamma \\not= 2$}\n\nIn this case, we have  \n\\begin{eqnarray}\n\\label{3.31}\nd{\\hat{u}}  &=& e^{au^{1-\\gamma\/2}}du,\\nonumber\\\\\n\\hat{h}   &=& \\frac{1}{2}a u^{1-\\gamma\/2} + \\alpha(u),\\nonumber\\\\\n\\hat{\\Phi}   &=& e^{b u^{1-\\gamma\/2}}, \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.31a}\na &\\equiv& \\frac{2\\epsilon_{a}}{2-\\gamma} \n\\left(\\frac{4\\omega^{2}d}{D+d-2}\\right)^{1\/2},\\nonumber\\\\\nb &\\equiv& - \\frac{2\\epsilon_{a}}{2-\\gamma} \n\\left(\\frac{\\omega^{2}(D-2)^{2}}{(D+d-2)d}\\right)^{1\/2}.\n\\end{eqnarray}\nThen, the non-vanishing frame components of the Riemann tensor are given by\n\\begin{eqnarray}\n\\label{3.33a}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n\\hat{\\gamma}_{0}^{2} e^{-2au^{1-\\gamma\/2}}\\nonumber\\\\\n& & \\times \\left\\{\n\\frac{a\\gamma(2-\\gamma)}{8 u^{\\gamma\/2+1}}\n+ \\frac{a^{2} (2-\\gamma)^{2} + 16\\omega^{2}}{16 u^{\\gamma}}\\right\\}\n\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n\\hat{\\gamma}_{0}^{2} \\frac{b(2-\\gamma)}{4}e^{-2au^{1-\\gamma\/2}}\\nonumber\\\\\n& & \\times \\left\\{\n\\frac{ \\gamma }{u^{\\gamma\/2+1}}\n- \\frac{(b-a) (2-\\gamma)}{u^{\\gamma}}\\right\\}\n\\delta^{a}_{b}.\n\\end{eqnarray}\nTherefore, for the choice $\\epsilon_{a} = +1$ we have\n\\begin{eqnarray}\n\\label{3.36}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} &=& \\cases\n\\infty, & $\\gamma> 0$,\\cr\n{\\mbox{constant}}, & $\\gamma = 0$,\\cr\n\\infty, & $- 2 < \\gamma < 0$,\\cr\n{\\mbox{finite}}, & $\\gamma \\le -2$,\\cr}\\\\\n\\label{3.36a}\n\\hat{\\theta}_{n} &=& \\cases\n\\infty, & $\\gamma> 2$,\\cr\n{(D-2)\\alpha'(0)}, & $\\gamma \\le 2$,\\cr}\n\\end{eqnarray}\nas $u \\rightarrow 0$, but now with $A, B = i, a$. Thus, the\ntidal forces  experiencing by a free-falling observer remain finite\nin the string frame at $u = 0$ for all the cases, except for the ones \nwhere $0 < \\gamma$ or $ -2 < \\gamma < 0$. As a result,\nthe spacetime is singular at $u = 0$ for these latter solutions.\nHowever, the singularity is weak for $|\\gamma| < 2$, because the distortion exerting on the \nobserver is still finite,\n\\begin{eqnarray}\n\\label{3.37}\n \\int{d\\lambda \\int{\\hat{R}^{(A)}_{\\;\\;\\;\\; (0)(B)(0)} d\\lambda}} \n &\\sim & A_{1} \\lambda^{2-\\gamma} + A_{2} \\lambda^{1-\\gamma\/2} \\nonumber\\\\\n&\\sim &  {\\mbox{finite}},\n\\end{eqnarray}  \nas $ \\lambda \\rightarrow 0 \\; (\\mbox{or} \\; u \\rightarrow 0)$ when\n$|\\gamma| < 2$, where $A_{1}$ and $A_{2}$ \nare finite constants. \n\nTherefore, for the choice $\\epsilon_{a} = +1$ the strong singularities of \nthe solutions with $\\gamma > 2$ at $u = 0$ in the Einstein frame now \nremain in the string  frame. The singularities of the solutions with $0 < \\gamma < 2$ \nare weak in both of the two frames. The solutions with $ -2 < \\gamma < 0$ \nis free from singularity in the Einstein frame, while they become singular \nin the string frame, although the nature of the singularities is still weak. \nThe solutions with $ \\gamma = 0$ and $\\gamma \\le -2$ are free from singularity \nat $u = 0$ in both of the two frames.  \n\nNote that for $\\gamma = 0$ Eq.(\\ref{3.33a}) shows that \n\\begin{equation}\n\\label{3.37a}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} \\rightarrow 0,\n\\end{equation}\nas   $u \\rightarrow \\infty$. Thus, in this case the spacetime singularity \nat $u = \\infty$ appearing in the Einstein frame now disappears in the \n$(D+d)$-dimensional string frame, although the null infinity $u = - \\infty$ \nstill remains singular [cf. Fig. \\ref{fig1a}].\n \n\nWhen $\\epsilon_{a} = - 1$, from Eq.(\\ref{3.33}) we find that\n\\begin{equation}\n\\label{3.38}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} = \\cases{0, & $\\gamma > 2$,\\cr\n\\infty, & $ 2 > \\gamma > 0$,\\cr\n{\\mbox{constant}}, & $\\gamma = 0$,\\cr\n\\infty, & $- 2 < \\gamma < 0$,\\cr\n{\\mbox{finite}}, & $\\gamma \\le -2$,\\cr}\n\\end{equation}\nas $u \\rightarrow 0$. It can be shown that in this case the nature of the \nsingularities of the solutions remains the same in both of the two frames\nfor $2 > \\gamma \\ge  0$ and $\\gamma \\le -2$, that is, in both frames\nit is  \nweak for $ 0 < \\gamma < 2$, and free of singularities for  $\\gamma = 0$\nand $\\gamma \\le -2$.\nFor $-2 < \\gamma < 0$, the solutions are free of singularities  \nin the Einstein frame, but singular in the string frame with the nature\nof the singularities being still weak.     \nFor $ 2 < \\gamma$, on the other hand, the solutions are free of singularities  \nin the string frame, but singular in the Einstein frame with the nature\nof the singularities being   strong.   \n\nSimilarly, one can show that for $\\gamma = 0$ the spacetime singularity \nat $u = -\\infty$ appearing in the Einstein frame now disappears in the \n$(D+d)$-dimensional string frame, although the spacetime is still\nsingular at the null infinity $u = + \\infty$ [cf. Fig. \\ref{fig1a}].\n \n\n\\subsubsection{$\\gamma = 2$}\n\nWhen $\\gamma = 2$, the corresponding solutions in the Einstein frame\nare given by Eqs.(\\ref{3.15}) and (\\ref{3.16}). The solutions have\na strong singularity at $u = 0$. In the string frame, the corresponding\nsolutions are given by Eq.(\\ref{3.30}) but with\n\\begin{eqnarray}\n\\label{3.39}\n\\hat{h}(\\hat{u}) & =& \\frac{a+2\\delta}{2(1+a)}\\ln\\left|\\hat{u}\\right|,\\nonumber\\\\\n\\hat{\\Phi}(\\hat{u}) & =& \\left((1+a)\\hat{u}\\right)^{\\frac{b}{1+a}}, \n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{3.39a}\n\\hat{u} \\equiv \\frac{1}{1+a} u^{1+a} = \\cases{0, & $ a > -1$,\\cr  \n- \\infty, & $ a < -1$,\\cr}\n\\end{equation}\nas $u \\rightarrow 0$. Thus, when $a > -1$ the half plane $u \\ge 0$ is mapped\nto the half plane $\\hat{u} \\ge 0$, and the hypersurface $u = 0 \\; (u = \n\\infty)$ is mapped to the one $\\hat{u} = 0 \\; (\\hat{u} = \n\\infty)$. when $a < -1$ the half plane $u \\ge 0$ is mapped\nto the one $\\hat{u} \\le 0$, and the hypersurface $u = 0 \\; (u = \n\\infty)$ corresponds to the one $\\hat{u} = -\\infty \\; (\\hat{u} = \n0)$. \n\nIt can be shown that now we have  \n\\begin{eqnarray}\n\\label{3.40}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n- \\hat{\\gamma}_{0}^{2} \\frac{(a+2\\delta)(2\\delta - a -2)}\n{4(1+a)^{2}\\hat{u}^{2}}\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2} \\frac{b(b - a -1)}\n{(1+a)^{2}\\hat{u}^{2}} \\delta^{a}_{b}.\n\\end{eqnarray}\nClearly, the spacetime is singular at $\\hat{u} = 0$, and the nature\nof the singularity is strong, because\n\\begin{equation}\n\\label{3.41}\n\\int{d\\lambda\\int{\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)}d\\lambda}} \\sim \\ln{\\lambda}\n\\rightarrow - \\infty,\n\\end{equation}\nas $ \\lambda \\rightarrow 0 \\; ({\\mbox{or}}\\; \\hat{u} \\rightarrow 0)$. \nNote that the distortion also becomes unbound as $|\\hat{u}| \\rightarrow    \\infty\n\\; (|\\lambda| \\rightarrow \\infty)$, although the tidal forces vanish there.\n \nIt should be noted that the above analysis is valid only for $a \\not= -1$.\nWhen $a = -1$, we find that\n\\begin{equation}\n\\label{3.42}\n\\omega^{2} = \\frac{D+d-2}{4d}, \\;\\; \\; \nb = \\frac{D-2}{2d}. \n\\end{equation}\nThe corresponding solutions are given by\n\\begin{eqnarray}\n\\label{3.43}\n\\hat{h}(\\hat{u}) &=& \\left(\\delta - \\frac{1}{2}\\right)\\hat{u},\\nonumber\\\\\n\\hat{\\Phi}(\\hat{u}) &=& e^{\\frac{D-2}{2d} \\hat{u}},\n\\end{eqnarray}\nwhere $u \\equiv  e^{\\hat{u}}$.\nThen, we find that\n\\begin{eqnarray}\n\\label{3.44}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n- \\hat{\\gamma}_{0}^{2} \\left(\\delta - \\frac{1}{2}\\right)^{2}\\delta^{i}_{j}, \\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2} \\left(\\frac{D-2}{2d}\\right)^{2}  \n\\delta^{a}_{b},\n\\end{eqnarray}\nwhich are finite (constants). However, at the null infinities \n$\\hat{u} = \\pm \\infty$, which correspond, respectively,  to $u = 0$ and $u = \\infty$,\nthe distortions are still unbound. As a result, the ($D+d$)-dimensional spacetimes  remain\nsingular on these surfaces. \n\nTherefore, when $\\gamma = 2$   the corresponding Penrose daigram of the ($D+d$)-dimensional \nspacetimes is that of Fig. \\ref{fig1a}, where the two hypersurfaces $u = 0$ and $u = \\infty$ \nremain singular.\n\n\n\n\n\n\n\\subsection{ $F'(u) G'(v)  \\not= 0$}\n\nIn this case, three classes of solutions were studied in the last section.  To have them \nmanagible, in this subsection we shall restrict ourselves only to $D = d = 5$. We generalize \nthe metric (\\ref{3.30}) and using $t=u+v$, $y=u-v$ we arrive at the form:\n\\begin{eqnarray}\n\\label{3.45}\nd{\\hat{s}}^{2}_{10} &=& \\frac{1}{2} e^{2A(t,y)} \\left(dt^2-dy^2\\right) - e^{2B(t,y)}d\\Sigma^{2}_{3} \\nonumber\\\\\n& &  - e^{2C(t,y)}d\\Sigma^{2}_{5,z},\n\\end{eqnarray}\nwhere $d\\Sigma^{2}_{5,z} \\equiv {\\hat{\\gamma}_{ab}\\left(z^{c}\\right)dz^{a}dz^{b}}\\;\n(a, b = 1, 2, ..., 5)$, and\n\\begin{eqnarray}\n\\label{3.46}\nA &=& \\sigma - \\frac{5}{3}\\beta \\phi = - \\frac{1}{3} \\ln(t) + \\kappa^{2}_{5}M -  \\frac{5}{3}\\beta \\phi,\\nonumber\\\\\nB &=& h - \\frac{5}{3}\\beta \\phi = \\frac{1}{3} \\ln(t) - \\frac{5}{3}\\beta \\phi,\\nonumber\\\\\nC &=&  \\beta \\phi, \\hspace{14pt}\n\\beta =  \\epsilon \\sqrt{\\frac{3\\kappa_{5}^{2}}{40}}, \\;\\;\\;\n\\epsilon_{a} = \\pm 1. \n\\end{eqnarray}\n\n\n\\subsubsection{Class IIa Solutions}\n\n\nIn this case, substituting the solution (\\ref{3.16af}) into Eq.(\\ref{3.46}) and setting \n$M_0=\\phi_0=0$ without loss of generality, we obtain that\n\\begin{eqnarray}\n\\label{3.47a}\nA  &=& \\frac{1}{2} \\left[\\left( c\\kappa_5 - \\epsilon \\sqrt{\\frac{5}{24}} \\right)^2  \n- \\frac{7}{8}\\right] \\ln(t), \\nonumber\\\\\nB  &=& \\left(\\frac{1}{3} - \\epsilon \\sqrt{\\frac{5}{24}} c\\kappa_5\\right) \\ln(t), \\nonumber\\\\\nC  &=& \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln(t).\n\\end{eqnarray}\nThe corresponding Kretschmann scalar is given by,\n\\begin{equation}\n\\label{3.48a}\nI_{10} \\equiv  R_{ABCD}R^{ABCD} = \\frac{\\tilde{I}_{10}}{t^{\\alpha_0}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\label{alpha0}\n\\alpha_0    &=& 2\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}} \\right)^2 + \\frac{9}{4} > 0, \\nonumber\\\\\n\\tilde{I}_{10} &=& \\frac{1}{45} \\left[9 \\chi^{2} \\left(40\\chi + 143\\right)  \n\t- \\epsilon_{a} 52 \\sqrt{30} \\chi^{3\/2}  \\left(3 \\chi + 4\\right)\\right.\\nonumber\\\\\n         & & \\left. + 80\\left(5 \\chi +2\\right)\\right],\n\\end{eqnarray}\nnow with $\\chi \\equiv c^{2}\\kappa^{2}_{5}$.\nClearly, the spacetime is always singular at $t = 0$ for any given $c$, similar to that in \nthe 5-dimensional case. Therefore, in the present case, the spacetime singularity remains even\nafter lifted  from the effective 5-dimensional spacetime to the 10 dimensional bulk. \n\n\n\\subsubsection{Class IIb Solutions}\n\n\nIn this case, the combination of Eqs.(\\ref{3.16ah}) and (\\ref{3.46}) yields\n\\begin{eqnarray}\n\\label{3.47b}\nA &=& \\left[2\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}}\\right)^2 \n- \\frac{7}{16}\\right] \\ln(t) \\nonumber\\\\\n  & & - \\left[\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}}\\right)^2 \n  - \\frac{5}{96}\\right] \\ln \\left (y+\\sqrt{y^2-t^2} \\right) \\nonumber\\\\\n  & & - \\frac{1}{2} \\chi^2 \\ln \\left( y^2 - t^2 \\right), \\nonumber\\\\\nB &=&  \\left( \\frac{1}{3} - \\epsilon_{a} \\sqrt{\\frac{5}{6}} c\\kappa_5 \\right) \\ln(t) \\nonumber\\\\\n  & & - \\epsilon_{a} \\sqrt{ \\frac{5}{24} } c \\kappa_5  \\ln \\left( y+\\sqrt{y^2-t^2} \\right),  \\nonumber\\\\\nC &=& \\epsilon_{a} \\sqrt{\\frac{3}{10}} c\\kappa_5 \\ln(t) \\nonumber\\\\\n  & & - \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln\\left( y+\\sqrt{y^2-t^2}\\right),\n\\end{eqnarray}\nfor which we find that\n\\begin{equation}\n\\label{3.48b}\nI_{10} = \\frac{\\tilde{I}_{10}}{t^{\\alpha_{0}}(y^2-t^2)^{\\alpha_{1}}(y+\\sqrt{y^2-t^2})^{\\alpha_{2}}}, \n\\end{equation}\nwhere $\\alpha_{0}$ is given by Eq.(\\ref{alpha0}),\n\\begin{eqnarray}\n\\label{alpha1}\n\\alpha_{1} &\\equiv& 2 \\left(\\frac{3}{4} - \\chi\\right),\\nonumber\\\\\n\\alpha_{2} &\\equiv& \\frac{101}{24} - \\left(2 c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}}\\right)^2,\n\\end{eqnarray} \nand $\\tilde{I}_{10} = \\tilde{I}_{10}(t,y)$, which is non-zero for $t = 0$ and\n$y^{2} = t^{2}$, but its  expression is too complicated to give it here explicitly. \n\nFrom the above expression, it can be seen that the spacetime is always singular at $t = 0$, but the\nstrenght of the singularity for $y > 0$ and $y < 0$ is different, because when $t = 0$ we have\n$y+\\sqrt{y^2-t^2} = 0$ for $y \\le 0$ and $y+\\sqrt{y^2-t^2} \\not= 0$ for $y > 0$.\nIn particular, when  $t = 0$ and $y > 0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t=0,\\; y>0} \\simeq \\frac{\\tilde{I}_{10}}{t^{\\alpha_{0}}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\label{aa}\n\\tilde{I}_{10} \\Big|_{t=0} & = & \\frac{256}{45} y^8 \\bigg[9\\chi^2 \\left(160 \\chi + 143\\right) \\nonumber\\\\\n    & & \n    - \\epsilon_{a} 104 \\sqrt{30} \\chi^{3\/2} \\left(3 \\chi + 1\\right) \\nonumber\\\\\n    & & \n    + 10 \\left(10 \\chi + 1\\right)\\bigg].\n\\end{eqnarray}\nOn the other hand, when $t = 0$ and $y < 0$, we find that  \n\\begin{equation}\nI_{10} \\Big|_{t=0, \\; y<0} \\simeq \\frac{y^{\\alpha_{2} - 2\\alpha_{1}} \\tilde{I}_{10}}\n{t^{{32}\/{3}}}, \n\\end{equation}\nwhere $\\tilde{I}_{10}$ is still given by Eq.(\\ref{aa}).  \n\nEqs.(\\ref{3.48b}) and (\\ref{alpha1}) also show that  the spacetime is singular when \n$y^{2} - t^{2} = 0$ for $\\chi < 3\/4$,   \n\\begin{equation}\nI_{10(b)} \\Big|_{t^2=y^2} \\propto \\frac{\\tilde{I}_{10}}{(y^2-t^2)^{\\alpha_{1}}},\n\\end{equation}\nbut now with\n\\begin{eqnarray}\n\\label{4.33}\n\\tilde{I}_{10} \\Big|_{t^2=y^2} &=& \\pm\\frac{8}{15}t^7\\chi \\bigg[20(6\\chi^{2}-\\chi-1) \\nonumber\\\\\n                    & & - 13\\epsilon_{a} \\sqrt{30} c\\kappa_5(2\\chi^2-1)\\bigg].\n\\end{eqnarray}\nThe corresponding Penrose diagram for $\\chi < 3\/4$ is given by Fig. \\ref{fig2b}.\nIt is remarkable to note that the solutions with $ 1\/2 \\le \\chi < 3\/4$ is not singular in the \n5-dimensional effective theory, as shown explicitly in Sec. III.B.2. \n\nSince the solution in the 10-dimensional bulk is not singular across the hypersurfaces $y^{2}\n= t^{2}$ for $\\chi \\ge 3\/4$, one must extend the solutions beyond these surfaces. The\nextension is quite similar to the 5-dimensional case for the ones with $\\chi \\ge 1\/2$.\nIn particular, for  $3\/4 \\le \\chi < 1$, setting \n\\begin{equation}\n\\label{4.33a}\n\\bar{u} = (y + t)^{2n},\\;\\;\\;\n\\bar{v} = (y - t)^{2n},\n\\end{equation}\nwhere $n$ is given by Eq.(\\ref{3.16ak}), one can show that the coordinate singularity at $y^{2} = t^{2}$\ndisappears in terms of   $\\bar{u}$ and $ \\bar{v}$. Then, the Penrose diagram for the extended\nsolutions is given exactly by Fig. \\ref{fig2a}. \n\nWhen $\\chi \\ge 1$, the hypersurfaces $y^{2} = t^{2}$ already  represent the null infinities, and\nthe corresponding Penrose diagram is given by Fig. \\ref{fig2b}, but  now the hypersurfaces \n$y^{2} = t^{2}$, represent,\nrespectively, by the lines $0D$ and $0E$, are non-singular. \nThe two regions $I$ and $I'$ are physically disconnected.\n\n\n\n\n\\subsubsection{Class IIc Solutions}\n\n\n\nIn this case, from Eq.(\\ref{3.16am}) we find that\n\\begin{eqnarray}\n\\label{3.47c}\nA &=& -\\frac{1}{3} \\ln(t) - \\frac{1}{2} \\chi^2 \\ln(y^2-t^2)  \\nonumber\\\\\n  & & + \\left[\\left( c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}} \\right)^2  \n  - \\frac{5}{96}\\right] \\ln \\left( y + \\sqrt{y^2-t^2}\\right),\\nonumber\\\\\nB &=& \\frac{1}{3} \\ln(t) - \\epsilon_{a} \\sqrt{\\frac{5}{24}} c \\kappa_5 \\ln\\left(y+\\sqrt{y^2-t^2}\\right), \\nonumber\\\\\nC &=& \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln \\left( y+\\sqrt{y^2-t^2} \\right),\n\\end{eqnarray}\nand that \n\\begin{equation}\n\\label{3.48c}\nI_{10} = \\frac{\\tilde{I}_{10}}{t^{{8}\/{3}}(y^2-t^2)^{\\alpha_{1}}(y+\\sqrt{y^2-t^2})^{\\alpha_{3}}}, \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{alpha3}\n\\alpha_{3} \\equiv \\frac{91}{24} + \\left( 2 c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}} \\right)^2 > 0,\n\\end{equation}\nand $\\tilde{I}_{10} = \\tilde{I}_{10}(t,y)$  is non-zero and finite at $t = 0$, but too complicated\nto be written out here. \nFrom the above expressions we can see that the spacetime is   singular at $t = 0$.\nThe strength of the singularities once again depends on $y < 0$\nand $y > 0$. In partucular, when  $t = 0$ and $y >0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t = 0, y > 0} \\propto \\frac{\\tilde{I}_{10}}{t^{8\/3}},\n\\end{equation}\nwith $\\tilde{I}_{10} \\simeq   {512}y^7\/9$. But for   $t = 0$ and $y < 0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t=0, y<0} = \\frac{ \\tilde{I}_{10}(y-\\sqrt{y^2-t^2})^{\\alpha_{3}}}\n{t^{{8}\/{3}+2\\alpha_{3}}(y^2-t^2)^{\\alpha_{1}}}, \n\\end{equation}\nwhere  $\\alpha_{3} >0$, as shown by Eq.(\\ref{alpha3}). Therefore, we again have a spacetime \nsingularity at $t=0$ for $y < 0$, but with more singular strength.\n\nThe singular behavior of the spacetime along the hypersurfaces $y = \\pm t$ are similar to the last\ncase. In particular, it is singular  for $\\chi < 3\/4$,  and corresponding Penrose diagram   is given \nby Fig. \\ref{fig2b}. It is interesting to note again that the solutions with $ 1\/2 \\le \\chi < 3\/4$ \nis not singular in the 5-dimensional effective theory. \n\nWhen $1> \\chi \\ge 3\/4$   the corresponding solutions in the 10-dimensional bulk is not singular \nacross the hypersurfaces $y^{2} = t^{2}$, one must extend the solutions beyond these surfaces. The\nextension is quite similar to the last case by simply introducing two null coordinates, defined by\nEq.(\\ref{4.33a}).  Then, the Penrose diagram for the extended\nsolutions is given exactly by Fig. \\ref{fig2a}. \n\nWhen $\\chi \\ge 1$, the hypersurfaces $y^{2} = t^{2}$ already  represent the null infinities, and\nthe corresponding Penrose diagram is given by Fig. \\ref{fig2b}.\n\n \n \n\\section{Conclusions and Discussing Remarks}\n\nAccording to the singularity theorems \\cite{HE73}, the 4-dimensional spacetimes are \ngenerically singular, if  matter filled the spacetimes satisfies certain energy condition(s).\nIt is a common belief that new physics involved with quantum mechanics will play an important\nrole    when the spacetime curvature is very high. The new physics gives us hope that\nthe classical singularities might be finally removed. \n\nHowever, it was shown that certain \nsingularities can be also removed by simply passing to a higher-dimensional theory of gravity, \nfor which spacetime is only effectively four-dimensional below some compactification scale \n \\cite{GHT95,RMS07}. This is because when lifted to higher-dimensions, the conditions required\n by the singularity theorems of Penrose and Hawking do not hold any more. \n\nIn this paper, we have investigated this  problem, by first studying the local and global properties\nof the spacetimes in the low dimensional effective theory, and then lifting them to the corresponding\n high dimensional spacetimes. We have shown explicitly that spacetime\nsingularities may or may not remain after lifted to higher dimensions, depending on the particular\nsolutions considered. We have also found that there exist cases in which the spacetime singularities of low \ndimensional effective theory occur in different surfaces from those of high dimensional spacetimes. \n\n\n \n \n\\begin{acknowledgements}\n\nOne of the authors (AW) would like to express his gratitude to Jianxin Lu for organizing  \nthe advanced workshop, ``Dark Energy and Fundamental Theory,\"  Xidi, Anhui, China,\nMay 27 - June 5, 2010, supported by the Special Fund for Theoretical Physics from the \nNational Natural Science Foundation of China with grant No. 10947203. Part of this work \nwas done during the  Workshop. He would also like to thank the other ten participants for \nvaluable discussions. The work of AW was supported in part by DOE Grant, DE-FG02-10ER41692.\n\n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the 50's, Goeppert-Mayer~\\cite{mayer} and Haxel, Jensen, \nand Suess~\\cite{suess} were able to obtain a description of the nuclear magic\nnumbers by adding a strong spin-orbit interaction to a central mean field potential. This leads to the\ncreation of the so-called spin-orbit gaps with magic numbers 2, 8, 20, 28, 50, 82 and 126. Stable nuclei with magic number of protons and\/or neutrons were found to \nbe spherical, having a closed shell configuration. This simple view of indestructible magic numbers has been modified over the last 30 years due to new spectroscopic information on exotic nuclei such as $^{32}_{12}$Mg$_{20}$~\\cite{32Mg,32MgB}. A reduction of the spherical $N=20$ gap was invoked to explain the deformation of $^{32}_{12}$Mg where a gain in correlation energy, due to quadrupole excitations obtained when promoting nucleons across the shell gap, is larger than the loss in energy required to cross that shell gap. Therefore, if the spherical shell gap is reduced, and\/or the two orbits forming the gap are separated by two units of angular momentum, as for all spin-orbit magic numbers, the nuclear deformation is favored. These two conditions are fulfilled also in $^{42}$Si, which was found to be deformed~\\cite{beyhan, bazin}. One of the key nuclei to understand how deformation sets in for the N~=~28 isotones between the spherical $^{48}$Ca and $^{42}$Si is $^{44}$S. \nThis nucleus has been investigated using different experimental\n approaches, such as $\\beta$-decay~\\cite{sorlin,grevy1}, Coulomb excitation~\\cite{glas}, in-beam $\\gamma$-ray spectroscopy~\\cite{dora}, isomer studies~\\cite{grevy,force}, electron spectroscopy~\\cite{force} and the two proton knock-out reaction~\\cite{santiago}. The works of Ref.~\\cite{glas, dora} showed that $^{44}$S has a deformed ground state. \nThe study of the decay of the 0$^+_2$ level~\\cite{force} and the B(E2:~2$_1^+\\rightarrow$0$^+_1$) reduced transition probability~\\cite{glas} led to the conclusion, by means of shell model calculations, that $^{44}$S exhibits a prolate ground state and a 0$^+_2$ spherical isomeric level.\n In addition, shell-model calculations predicted the existence of a deformed band built on top of the ground state\n and a 2$^+_2$ level of spherical character above the 0$^+_2$ state. A candidate for the 2$^+_2$ level has been proposed in Ref.~\\cite{santiago}. The aim of the present work is to confirm the existence of this state, among others found in Ref.~\\cite{santiago}, and to discover new-high energy levels connecting the deformed and spherical bands in $^{44}$S. All this information is relevant to further understand shape~-~coexistence in this nucleus. The authors of Ref.~\\cite{santiago} used a two proton knock-out reaction from a secondary beam  of $^{46}$Ar to populate excited states in $^{44}$S while, in the present work, a one proton knock-out reaction $^{45}$Cl(-1p)$^{44}$S was employed. This different reaction mechanism is expected to populate additional states in $^{44}$S leading to a complementary understanding of the structure of this nucleus.\n\n\\section{Experimental method}\n\n\\begin{figure}\n\\includegraphics[width=0.50\\textwidth]{caceres_ID_rev.eps}\n\\caption{\\label{fig:z_aoq}(Color online) (a): Energy loss ($\\Delta$E) versus time of flight (TOF) particle identification plot of the ions produced in the fragmentation of $^{48}$Ca beam on C target and separated in the ALPHA spectrometer. (b): Energy loss ($\\Delta$E) versus A\/Q plot of the ions produced in the fragmentation of the $^{45}$Cl secondary beam in the Be secondary target and separated in the SPEG spectrometer.}\n\\end{figure}\n\nThe experiment was performed at the Grand Acc\\'el\\'erateur National d'Ions Lourds (GANIL, France) facility.\nA $^{48}$Ca$^{19+}$ primary beam at 60~MeV\/u and 3.8~$\\mu$A average intensity impinged on a 200~mg\/cm$^2$ C target located between the two superconductor solenoids of the SISSI~\\cite{sissi} device.\nThe reaction products were separated by means of the B$\\rho$-$\\Delta$E-B$\\rho$ method~\\cite{dufour, Anne} in the ALPHA\n spectrometer. The ion identification was performed on an event-by-event basis by measuring their time-of-flight (TOF) over an 80~m flight path using two micro-channel plates and energy loss ($\\Delta$E) in a 50~$\\mu$m Si detector located at the entrance of the SPEG spectrometer. An example of the particle identification plot for the reaction residues is shown in Fig.~\\ref{fig:z_aoq}~(a). \nAfterwards, the selected fragments underwent secondary reactions in a 195~mg\/cm$^2$ $^9$Be target placed at the Si detector position. The $\\Delta$E and positions of the secondary ions were measured at the final focal plane of SPEG in ionization and drift chambers, respectively. A plastic scintillator was used to determine the TOF and residual energy. All of these measurements permitted a full reconstruction of the mass-to-charge ratio of the produced secondary ions (Fig.~\\ref{fig:z_aoq}~(b)). \nAlthough, the magnetic rigidity of the spectrometer was optimized \nfor $^{42}$Si transmission, the large acceptance of SPEG ($\\Delta$p\/p$\\sim$7\\%) spectrometer permitted 50\\% of the $^{44}$S momentum distribution from the one proton knock-out reaction from $^{45}$Cl to be transmitted. This reaction is of particular interest as it provides spin and parity selectivity of the final states populated.\nThe inclusive $\\sigma_{1p}$($^{45}$Cl$\\rightarrow^{44}$S) cross section was measured to\n be 13(3)~mb. The prompt-$\\gamma$ rays detected in coincidence with the nuclei identified at the final focal plane of SPEG\n were measured in 70~BaF$_2$ detectors of the Ch\\^{a}teau de Cristal array positioned above and below \nthe secondary target at an average distance of 25~cm. The efficiency of the\n array was determined using standard sources of $^{60}$Co and $^{152}$Eu to be 24\\% at 1.33~MeV with an energy resolution of 15\\% at 800~keV. A time resolution of 800~ps was obtained. \n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{caceres_SPEC.eps\n\\caption{\\label{fig:energy} (a): Singles $\\gamma$-ray spectrum in coincidence with $^{44}$S ions.\n Coincidence spectra gated by the (b) 1321, (c) summed 977 and 1006, (d) 1979 and (e) 1198~keV transitions, respectively.}\n\\end{figure}\n\n\n\\section{results}\n\n\n\nThe $\\gamma$-ray singles spectrum in coincidence with $^{44}$S reaction products is shown in Fig.~\\ref{fig:energy}(a). Transitions previously reported in Ref.~\\cite{glas,dora} are confirmed with the exception of the $\\gamma$-ray at 2632~keV that is not visible in the present work. However, a broad $\\gamma$-ray distribution at high excitation energy is observed.\nThe construction of the level scheme shown in Fig.~\\ref{fig:ls} is based on a exhaustive $\\gamma\\gamma$-coincidence analysis described below. The fitting function employed to obtain the $\\gamma$-ray intensities and their energies has, as a fixed parameter, the energy-dependent peak-width calculated from the systematics of known transitions of other reaction channels measured in the same experiment.\n\n\\begin{figure}\n\\includegraphics[width=.50\\textwidth]{caceres_HE_rev.eps}\n\\caption{\\label{fig:fit} (Color online) High energy part of the spectra obtained (a) from the $\\gamma\\gamma$ matrix gated by the 1321~keV transition and (b) singles $\\gamma$-ray. The lines shows the result of the fit with two (a) and three (b) Gaussian functions, respectively. The p-value of the $\\chi^2$ goodness-of-fit test is represented by the letter P.}\n\\end{figure}\n\nThe spectrum obtained from the $\\gamma\\gamma$-matrix gated on the 2$^+_1\\rightarrow$0$^+_1$ 1321~keV $\\gamma$ transition is shown in Fig~\\ref{fig:energy}(b). Two well separated $\\gamma$ rays at high excitation energy are observed. A fit to the high~-~energy part of this spectrum with a two Gaussian functions yields 1979 and 2262~keV for the two $\\gamma$-rays, respectively (Fig.~\\ref{fig:fit} (a)). The p-value from the $\\chi^2$ goodness-of-fit test is 0.84.\nIt was not possible to fit the broad high energy distribution visible in the singles spectrum with two Gaussian functions if the 1979 and 2262~keV centroids are considered as fix parameters (p-value~$<$~0.07). Similarly, if the high energy part of the singles spectrum is fitted first with two Gaussian functions and then the energies obtained are used as a fixed parameter in the fitting procedure of the spectrum gated on the 1321~keV, the p-value of the latter fit yields 0.3. The energies of the two $\\gamma$ rays obtained by the two methods are thus not consistent.\nTherefore, this analysis suggests that the high energy and broad $\\gamma$-ray distribution visible in the single spectrum is actually composed of three different $\\gamma$-rays.  \nThe fit of this distribution using three Gaussian functions, in which the centroids and widths of the 1979 and 2262 keV transitions observed in Fig.~\\ref{fig:energy}(b) are fixed, yields 2156~keV as the energy of the third $\\gamma$-ray (Fig.~\\ref{fig:fit}(b)). These three transitions are visible with a significance level of more than 4~$\\sigma$. The 2156~keV $\\gamma$-ray is not in coincidence with the 1321~keV transition. Therefore, it is proposed that it decays directly from a level at 2156~keV to the ground state of $^{44}$S. The observation of a 1979~keV $\\gamma$ ray connected to the 1321~keV 2$^+_1$ state establishes a level at about 3300~keV excitation energy (see below).\nThe structure in the $\\gamma$-ray spectrum between 950 and 1050~keV is a doublet composed of two different transitions at 977 and 1006~keV [Fig~\\ref{fig:energy}(c)]. The sum of the 977 and 1006~keV $\\gamma$ energies leads to 1983(34)~keV which, within the experimental uncertainties, agrees with the energy of the 1979(19)~keV $\\gamma$ ray. Therefore, these three transitions most likely belong to two parallel branches that de-excites from a new state. This state decays by a direct 1979 keV $\\gamma$-ray and a cascade of 977 and 1006~keV transitions. The energy spacing between the proposed level and the 2$^+_1$ state at 1321~keV can be estimated using the weighted mean of the energies of the two decay branches and yields 3301(18)~keV. \n\nThe 977~keV $\\gamma$-ray is placed immediately above the 2$^+_1$ state based on the fact that its energy matches the one of 949(5)~keV reported in Ref.~\\cite{santiago}. The 1006~keV transition is placed above the 977~keV and this $\\gamma $-ray was not observed in Ref.~\\cite{santiago}. This establishes the level at 2298(25)~keV corresponding that of 2268(5)~keV from Ref.~\\cite{santiago}. \nThe spectra shown in Fig~\\ref{fig:energy}(b) and (c) reveal the possible existence of a new $\\gamma$ ray at 1198~keV. This transition is likely in coincidence with all other $\\gamma$ rays with the exception of the 2262 and 2156~keV peaks. It is placed above the 1979~keV $\\gamma$ line depopulating a state at 4499~keV excitation energy with spin and parity Y$^+$ (see below).\n\n\\begin{table*}\n\\caption{\\label{tab:inten} Experimental energies, tentative spin and parity assignments, relative intensities and transition energies for the excited states in $^{44}$S. The direct feeding to the measured levels have been normalized to the\n number of detected $^{44}$S in SPEG and was obtained considering the 2262~keV transition feeding the 1321~keV (D$_{A}$) or the (2$^+_3$) (D$_{B}$) state (see text). The width of the $\\gamma$-ray ($\\sigma$) used in the fitting function is shown in the last column.}\n\\begin{minipage}{10.3cm}\n \\begin{tabular}{cccccccccccccc}\n\\hline\n\\hline\nE$_i$ [keV] &I${_i^\\pi}$ $\\rightarrow$ I${_f^\\pi}$&E$_\\gamma$ [keV]& I$_\\gamma[\\%]$&D$_A$[\\%]&D$_B$[\\%]&$\\sigma$[keV]&&\\\\\n\\hline\n1321(10)&2$^+_1\\rightarrow$0$^+_1$&1321(10)&100(8)&14(6)&20(7)&69(1)&\\\\\n2156(49)&(2$^+_2$)$\\rightarrow$0$^+_1$&2156(49)&17(6)&12(4)&12(4)&99(1)&\\\\\n2298(25)&(2$^+_3$)$\\rightarrow$2$^+_1$&977(23)\\footnote[1]{Extracted from the $\\gamma\\gamma$ matrix gating on the 950~-~1050~keV distribution [Fig~\\ref{fig:energy}(c)].}&48(6)&25(4)&11(4)&57(1)&\\\\\n3301(18)&(2$^+_6$,1$^+_2$)$\\rightarrow$(2$^+_3$)&1006(25)$^b$&12(3)&12(2)&13(2)&58(1)&\\\\\n&(2$^+_6$,1$^+_2$)$\\rightarrow$(2$^+_1$)&1979(19)\\footnote[2]{Extracted from the $\\gamma\\gamma$ matrix gating on the 1321~keV transition [Fig~\\ref{fig:energy}(b)].}&24(5)&&&93(1)&\\\\\n3583(39)\\footnote[3]{State depopulated by the 2262~keV transition feeding the 2$^+_1$ level.};4560(45)\\footnote[4]{State depopulated by the 2262~keV transition feeding the (2$^+_3$) level. }&(X$^+$)$\\rightarrow$2$^+_1$;(2$^+_3$)&2262(38)$^c$&21(5)&15(3)&15(3)&103(1)&\\\\\n4499(37)&(Y$^+$)$\\rightarrow$(2$^+_6$,1$^+_2$)&1198(25)$^b$&18(3)&13(2)&13(2)&65(1)&\\\\\n\\hline\n\\hline\n\\end{tabular}\n \\end{minipage}\n\\end{table*}\n\n\nFrom the present data, it is not possible to place the 2262~keV $\\gamma$ ray in the level scheme. The coincidence relation with the 1321~keV line indicates that it must decay from a state above the 2$^+_1$ level, labeled X$^+$ in Table~\\ref{tab:inten}.\nThe broad structure observed around 200-300~keV in the spectrum of singles (Fig.\\ref{fig:energy}(a)) is not associated with $^{44}$S. This structure corresponds to a peak at constant energy in the non Doppler-corrected spectra and, consequently, must originate from a stopped source. As shown in  Fig.~\\ref{fig:ls}, the present level scheme agrees with the one of Ref.~\\cite{santiago} for the first three excited states. In addition, two new levels are proposed at higher energy in the present work that were not observed in Ref.~\\cite{santiago} and highlights the different selectivity of the complementary reactions used in the two experiments.\n\nThe direct feeding to the levels normalized to the total number of $^{44}$S detected in SPEG have been\n extracted (Tab.~\\ref{tab:inten}). They are based on the measured $\\gamma$-ray intensities and calculated branching ratios. As the precise location of the 2262~keV $\\gamma$-ray in the level scheme could not be determined unambiguously, the direct population to states in $^{44}$S were deduced considering two assumptions: (D$_A$) in which the 2262~keV transition feeds the 2$^+_1$ level or (D$_B$) in which the 2262~keV transition feeds the 2298~keV state. It is interesting to note that the direct feeding of all the states populated in $^{44}$S is nearly the same (between 10 and 20\\%). This feature is a direct consequence of the large configuration mixing between levels in $^{44}$S. In contrast, the fact that none of the newly observed transitions is in coincidence with the 2156~keV $\\gamma$ ray indicates a different configuration for the 2156~keV state.\n\n\\section{Discussion}\n\nThe tentative spin and parity assignment to the observed levels in $^{44}$S is based on the comparison between the calculated and experimental (i) excitation energies, (ii) branching ratios and (iii) direct population of the states following the one proton knock-out reaction. \nShell model calculations (Fig.~\\ref{fig:ls}) using the SDPF-U interaction~\\cite{SDPF},\n known to reproduce the spectroscopic information in the region, were performed. The full model space was considered,\n comprising the proton $\\pi$(d$_{5\/2}$, d$_{3\/2}$, s$_{1\/2}$) and neutron $\\nu$(f$_{7\/2}$, p$_{3\/2}$, p$_{1\/2}$,\n f$_{5\/2}$) orbitals. Calculations were performed with the ANTOINE code~\\cite{antoine1,antoine2}. $E2$ reduced transition probabilities were calculated using polarization charges of 0.35~$e$ for both protons\n and neutrons. Effective magnetic moments for the $M1$ transition rate calculations are taken from\n Ref.~\\cite{SM} and were derived from systematic studies in the $sd$ shell. They correspond to the bare values quenched by 0.75 for the spin operator while the orbital operator is set 1.1 and -0.1 for proton ($g_p$) and neutron ($g_n$) gyromagnetic moments, respectively. These values ensure a very good agreement with the measured g-factors in $^{44}$Cl~\\cite{rydt} \nand $^{43}$S~\\cite{43S}. \n\n\\begin{figure*}\n\\includegraphics[width=1.\\textwidth,height=100mm]{caceres_LS_rev.eps}\n\\caption{\\label{fig:ls}Experimental level scheme of $^{44}$S obtained from (a) Force:~Ref.~\\cite{force}, (b) Santiago-Gonzalez:~Ref.~\\cite{santiago} and (c) in this work. The width of the arrows in (c) indicates the relative $\\gamma$-ray intensities normalized to the 1321~keV transition. The energy uncertainties on the $\\gamma$-rays and levels are quoted in Table~\\ref{tab:inten}. (d) Shell model calculations performed using the SDPF-U interaction. The arrow length in the right hand side of the figure shows the error on the neutron separation energy (S$_n$~=~5220(440)~keV~\\cite{Sn}). The levels observed in this data are represented with thicker widths line.}\n\\end{figure*}\n\n\nTheoretical direct population of the levels in $^{44}$S has been obtained combining the eikonal-model calculations~\\cite{tostevin} for the single particle cross sections with the shell model spectroscopic factors (SF) following the prescription of Ref.~\\cite{gade08}. The ground state spin and parity of $^{45}$Cl are not firmly known experimentally~\\cite{sorlin2,gade2}. Shell model calculations predict an I$^\\pi$=1\/2$^+$ assignment although the 3\/2$^{+}$ level is calculated to be at only 132~keV excitation energy. Therefore, both spins (1\/2 or 3\/2) were considered in the SF calculations (Fig.~\\ref{fig:SF}). The range of possible states populated in $^{44}$S from the proton knock-out reaction of $^{45}$Cl are from 0$^{+}$~to~3$^{+}$ (0$^{+}$~to~4$^{+}$) assuming 1\/2$^+$ (3\/2$^+$) for the ground state of $^{45}$Cl. Therefore, all calculations were restricted to these spin states~(Fig.~\\ref{fig:ls}). \n\n\nThe calculated 2$^+_1$ level at 1172~keV excitation energy corresponds to the 1321~keV experimental state previously reported in Ref.~\\cite{glas}. \nBetween 2000 and 2800~keV five levels were calculated. Among them, only the 2$^+_2$ state at 2140~keV excitation energy has a spherical configuration with intrinsic quadrupole moment -0.3~efm$^2$~\\cite{force}. It is predicted to decay by 83\\% and 16\\% to the 0$^+_1$ and 2$^+_1$ levels, respectively. Its decay to the 0$^+_2$ level, although favored in terms of reduced transition probability (B(E2:~2$_2^+\\rightarrow$0$^+_2$)$\\simeq$~3.2~B(E2:~2$_2^+\\rightarrow$0$^+_1$)), amounts to only 1\\%. This is due to the energy dependence of the transition probabilities. Although, having comparable configurations, the 2$^+_2$ and 0$^+_2$ are thus only weakly connected to each other. \nThe experimental analogue of the 2140~keV state is the 2156~keV level and the non observation of the 835~keV $\\gamma$ ray associated with the 2$^+_2\\rightarrow$2$^+_1$ decay is in agreement with experimental results considering the intensity of the 2156~keV $\\gamma$-ray. The 2156~keV state is tentatively assigned to be the (2$^+_2$) level of spherical character based on the fact that if this state had had a more mixed configuration, higher lying states would have decayed through it and been detected in this experiment. \nAs shown in Fig.~\\ref{fig:SF}, the calculated direct population to the 2$^+_2$ state is 5\\% and 1\\% assuming 1\/2$^+$ and 3\/2$^+$ spin and parity assignment for the ground state of $^{45}$Cl, respectively. The theoretical direct feeding to the 4$^+_1$ state (calculated to be 2569~keV and observed at 2447(5)~keV in Ref.~\\cite{santiago}) is 0\\% in the case of a 1\/2$^+$ and 2\\% in the case of a 3\/2$^+$ spin and parity of the ground state of $^{45}$Cl, respectively. Both, the experimental feeding of 12~\\% to the 2$^+_2$ and the non observed feeding to the 4$^+_1$ state rather favor a I$^\\pi$~=~1\/2$^+$ ground state for $^{45}$Cl.\nThe other four calculated states between 2000 and 2800~keV (i.e. 2$^+_3$, 3$^+_1$, 4$^+_1$, 4$^+_2$) are predicted to decay mostly to the 2$^+_1$ level ($>$90\\%); only the 2$^+_3$ state is expected to be strongly populated in the reaction (Fig.~\\ref{fig:SF}), independent of the I$^\\pi$ value of the ground state of $^{45}$Cl. For this reason, an I$^\\pi$~=~(2$^+_3$) spin and parity is assigned to the 2298~keV level. Above 3000~keV there is a large level density is predicted. A study of the decay modes of each state was performed considering the theoretical reduced transition probabilities and the experimental excitation energies (when available), otherwise they were taken from theory. In the latter case, the branching ratio of each level was investigated considering a variation of 500~keV of the theoretical excitation energies. Only the 1$^+_2$ and 2$^+_6$ states decay by competing and parallel branches to the 2$^+_3$ and 2$^+_1$ levels in agreement with the experimental data.!\n  Therefore, both states are possible candidates to spin and parity assignment for the 3301~keV level. In addition, the theoretical cross section calculations further support the I$^\\pi$~=~(1$^+_2$, 2$^+_6$) assignment (Fig.~\\ref{fig:SF}). Due to the large number of calculated levels around 4000~keV, it was not possible to find a theoretical counterpart to the experimental 4499~keV level, therefore no spin assignment can be given.\n\n\\begin{figure}\n\\includegraphics[width=.550\\textwidth,height=60mm]{caceres_SF_rev.eps}\n\\caption{\\label{fig:SF} (Color online) Theoretical partial cross sections to the levels in $^{44}$S normalized to the strength below the neutron separation energy S$_n$~=~5220(440)~keV assuming 3\/2$^+$ (a) and  1\/2$^+$ (b) spin and parity assignment for the ground state of $^{45}$Cl.}\n\\end{figure}\n\n\n\\section{Summary}\nThe $^{44}$S nucleus has been investigated using in-beam $\\gamma$-ray spectroscopy following the one proton knock-out reaction from $^{45}$Cl. The large efficiency of the BaF$_2$ detectors of the Ch\\^{a}teau de Cristal array, together with the particle selection provided by the SPEG spectrometer, allowed the construction of the level scheme up to high excitation energy. Analysis of $\\gamma\\gamma$-coincidences relations have been realized in this nucleus and the experimental direct feedings of the states were extracted. State-of-the-art nuclear shell model calculations have been performed for comparison to experimental energies, deduced spin assignments and decay branches of the states, and estimate their direct feeding from the one proton knock-out reaction. The energies and tentative spin assignments of the three excited states (2$^+_1$, 2$^+_2$ and 2$^+_3$) determined in Refs.~\\cite{force,santiago} are confirmed. The 2$^+_2$ level at 2156(49)~keV is found to have a specifi!\n c configuration, and is likely spherical. This is based on the weak feeding observed from other excited states, its feeding intensity from the (-1p) knock-out reaction and its decay pattern.\nTwo new excited states have been proposed at higher excitation energy with the 3301~keV level tentatively assigned to have I$^\\pi$~=~(1$^+_2$, 2$^+_6$), while no spin and parity could be determined for the level at 4499~keV excitation energy. \nThe information from  the present data, and the one reported in Ref.~\\cite{force,santiago}, highlights the large density of states at low excitation energy in the semi-magic $^{44}$S nucleus. This confirms the complexity of its nuclear structure that combines both shape and configuration coexistence.\n\n\n\\begin{acknowledgments}\nThis work benefits from discussions with G.~F.~Grinyer. \nThe authors are thankful to the\nGANIL and LPC staffs. We thank support by BMBF\nNo. 06BN109, GA of Czech Republic No. 202\/040791,\nMEC-DGI-(Spain) No. BFM2003-1153, and by the EC\nthrough the Eurons Contract No. RII3-CT-3\/2004-506065\nand OTKA No.~K68801 and Bolyai J\\'anos Foundation. R. B., A. B.,\nM. L., S. I., F. N. and R. C., X. L. acknowledge the\nIN2P3\/CNRS and EPSRC support.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\\setcounter{page}{1}\n\\section{Motivation}\nRecent Artificial Intelligence (AI) research has shown considerable interest in developing systems that can mimic human learning and thinking  \\cite{motivationintro}. Similar to human learning, there has been significant success in transferring knowledge between AI systems, like a ``student\" - ``teacher\" relationship \\cite{generalkt}. To simulate human learning, a ``student\" system should also be able to accumulate knowledge and combine the obtained information. For example, it is not expected of a Computer Science pupil to learn how to implement quicksort without first understanding the notions of median and partition. We can extend this expectation to an AI system. Instead of solving a complex task without prior knowledge, firstly, learn how to resolve more straightforward problems from the same area. This form of Machine Learning is referred to as multi-task learning \\cite{zhang2018overview}.\n\nInductive Logic Programming is an ideal candidate for the above use case \\cite{ILPat30}.\nInductive Logic Programming (ILP) is a form of machine learning that learns logic programs (also referred to as symbolic programs) based on given examples and background knowledge (BK). The BK set consists of a set of logic programs given by the user. Therefore, transferring knowledge becomes trivial \\cite{johnthesis}: if we want to reuse a solution in solving another task, it is sufficient to append it to the given BK. Returning to the ``student\" - ``teacher\" example, the previously mentioned approach would be equivalent to having a teacher provide his student with a new definition.\n\nAside from providing support for multi-task learning, ILP has been shown to provide extensive benefits compared to most ML models, such as Data Efficiency (being able to generalise from a small number of examples) and Explainability (providing outputs that are easily understandable by a human) \\cite{ILPat30}. \n\nWe will proceed by providing the reader with an example logic program (see \\autoref{fig:ILPexample}) written in Prolog in order to accommodate them with the syntax used throughout the project. Afterwards, we will describe four problems that can be solved by an ILP system and which would benefit from multi-task learning.\n\\begin{figure}[hbt!]\n    \\begin{minted}{prolog}\n   \n        isFather(\"John\", \"Dan\")\n        isFather(\"Paul\", \"John\")\n        isWife(\"Alice\", \"Paul\")\n        \n        isGrandmother(X,Y):-isWife(X,Z),\n                            isGrandfather(Z,Y)\n                                              \n                                              \n                                                \n        isGrandfather(X,Y):-isFather(X,Z),\n                            isFather(Z,Y) \n                                          \n                                          \n                                            \n   \n        ?-isGrandmother(\"Alice\",\"Dan\")\n    \\end{minted}\n    \\caption{Logic program example}\n    \\label{fig:ILPexample}\n\\end{figure}\n\n\\paragraph{Kinship relationships.} Kinship relationships allow an illustration of the problem solved by introducing multi-task learning in ILP systems. \nMoreover, the trivial nature of the problem allows for synthetic data generation to benchmark. In \\autoref{fig:kinship_example} we show a situation where a shorter program (\\texttt{isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)} instead of \\texttt{isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)}) is learnt to explain the grandmother relationship. \n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)\n    \\end{minted}\n    \\caption{Example for the kinship relationship problem}\n    \\label{fig:kinship_example}\n\\end{figure} \n\n\\paragraph{String transformation.} One of the most widely used examples to benchmark ILP systems are EXCEL type string transformation operations \\cite{ILPat30}. \nThrough multi-task learning, an ILP system can better capture situations where a complex transformation consists of sequential applications of more simple operations. In \\autoref{fig:string_example}, we present our initial set of example and their equivalent programs. The given functions perform the following operations:\n\\begin{itemize}\n    \\item \\texttt{f0} - capitalize each word of a string\n    \\item \\texttt{f1} - capitalize a single word\n    \\item \\texttt{f2} - drop the first word of a string and capitalize the second\n    \\item \\texttt{f3} - copy a string until the first non-letter character is encountered\n\\end{itemize}\nThe predicates \\texttt{inv0} and \\texttt{inv2} from the example are auxiliary functions invented during the learning process.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        f0('gerson zaverucha', 'Gerson Zaverucha')\n        f0('jesse davis', 'Jesse Davis')\n        f0('paolo frasconi', 'Paolo Frasconi')\n        \n        f1('james', 'James')\n        f1('jozie', 'Jozie')\n        f1('paul', 'Paul')\n        \n        f2('dr wilson', 'Wilson')\n        f2('dr brown', 'Brown')\n        f2('dr wright', 'Wright')\n        \n        f3('artificial.', 'artificial')\n        f3('intelligence ', 'intelligence')\n        f3('systems1a', 'systems')\n\n   \n        f0(A,B):-f1(A,C),inv0(C,B).\n        inv0(A,B):-copyskip1(A,C),f1(C,B).\n        \n        f1(A,B):-mk_uppercase(A,C),f3(C,B).\n        \n        f2(A,B):-skip1(A,C),inv2(C,B).\n        inv2(A,B):-skip1(A,C),f1(C,B).\n        inv2(A,B):-skip1(A,C),inv2(C,B).\n        \n        f3(A,B):-copyskip1(A,B),not_letter(B).\n        f3(A,B):-copyskip1(A,C),f3(C,B).\n    \\end{minted}\n    \\caption{Example for the string transformation problem}\n    \\label{fig:string_example}\n\\end{figure} \n\\paragraph{Robot movement.} Robot movement can be represented as basic movements such as going up\/down\/left\/right on a grid. For traversing large distances or completing complex paths it would be more efficient to have common patterns such as corner turns, as part of the background knowledge. \n\nIn \\autoref{fig:robot_example}, we show a solution where a 5 step translation is reduced to only two literals.\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        f(State(X, Y, Xnew, Ynew, Width, Height))\n        \n   \n        turn(State(0, 1, 1, 2, 5, 2))\n        turn(State(0, 0, 1, 1, 5, 2))\n        \n        jump(State(0, 1, 1, 4, 5, 2))\n        jump(State(0, 0, 0, 3, 5, 2))\n        \n        turnAndJump(State(0, 0, 1, 4, 5, 2))\n\n   \n        turn(State(X, Y, Xnew, Ynew, W, H)) :- right(X, Y, X2, Y2),\n                                               down(X2, Y2, Xnew, Ynew)\n                                                    \n        jump(State(X, Y, Xnew, Ynew, W, H)) :- right(X, Y, X2, Y2),\n                                               right(X2, Y2, X3, Y3)\n                                               right(X3, Y3, Xnew, Ynew)\n                                                        \n        turnAndJump(State(X, Y, Xnew, Ynew, W, H)) :- \n                                                turn(X, Y, X2, Y2),\n                                                jump(X2, Y2, Xnew, Ynew)\n    \\end{minted}\n    \\caption{Example for the robot movement problem}\n    \\label{fig:robot_example}\n\\end{figure} \n\n\\paragraph{Drawing images.} Most graphic rendering techniques are performed by decomposing scenes into primitives such as lines, triangles or quads. Specific algorithms (such as Constructive Solid Geometry) also make use of more complex shapes, for which the primitive decomposition is computed in advance \\cite{Graphics21}.\n\nBased on the above, we formulate the following problem: given a set of functions that control a printer head (moving over the four axes, returning to (1,1) coordinates or drawing a pixel) the system has to learn to draw complex images. In \\autoref{fig:drawing-images} we show how a cube can be drawn by connecting four lines.\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n   \n        f(State(X, Y, Width, Height, Grid))\n        \n   \n        topLine(State(_, _, 3, 3, [1,1,1,0,0,0,0,0,0]))\n        bottomLine(State(_, _, 3, 3, [0,0,0,0,0,0,1,1,1]))\n        leftLine(State(_, _, 3, 3, [1,0,0,1,0,0,1,0,0]))\n        rightLine(State(_, _, 3, 3, [0,0,1,0,0,1,0,0,1]))\n        isCube(State(_, _, 3, 3, [1,1,1,1,0,1,1,1,1]))\n        \n   \n        topLine(S, S5) :- at_start(S), move_up(S, S2), move_up(S, S3), \n                             draw1(S3), move_right(S3, S4), draw1(S4), \n                             move_right(S4, S5), draw1(S5)\n        bottomLine(S, S3) :- at_start(S), draw1(S), move_right(S, S2), \n                                draw1(S2), move_right(S2, S3) draw1(S3)\n        leftLine(S, S3) :- at_start(S), draw1(S), move_up(S, S2), \n                              draw1(S2), move_up(S2, S3) draw1(S3)\n        rightLine(S, S5) :- at_start(S), move_right(S, S2), \n                               move_right(S, S3), draw1(S3), \n                               move_up(S3, S4), draw1(S4), \n                               move_up(S4, S5), draw1(S5)\n        isCube(S, S4) :- topLine(S, S2), returnStart(S2, S3), \n                         bottomLine(S3, S4), returnStart(S4, S5),\n                         leftLine(S5, S6), returnStart(S6, S7),\n                         rightLine(S7, S8), returnStart(S8, S9)\n    \\end{minted}\n    \\caption{Example for the image drawing problem}\n    \\label{fig:drawing-images}\n\\end{figure}\n\nAll the above problems are solvable by an ILP system, but certain limitations exist. ILP systems learn by searching through a hypothesis space of possible solutions, but even the most advanced search techniques face difficulties when the size of the hypothesis space increases significantly. The size of the hypothesis space is directly related to the maximum size a program can have \\cite{popper}. In order to better understand this statement, imagine the task of writing down all possible programs of size $n$ that contain a given list of predicates (allowing duplicates). One can easily observe that the size of the result grows exponentially with $n$. Therefore, the maximum size of possible solutions must be restricted such that searching through it is feasible. Consequently, if the solution for a task is too large, the task will become unsolvable. \n\nMulti-task learning mitigates the above issue by reusing learnt solutions when learning tasks from related domains. In the four previous examples, we can observe that several tasks can be represented as more compact solutions by reusing the solutions from some other tasks.\n\nThe state-of-the-art approach to ILP multi-task learning is based on the Metagol \\cite{metagol} system and was described by \\citet{lin2014bias} in 2014. It has been shown to approach the performance of commercial systems of the time and outperform an independent learning strategy (running Metagol on each task separately). In \\autoref{implementation} we will identify significant drawbacks of the current strategy that negatively impact its learning abilities. \n\nFinally, we introduce Popper \\cite{popper}, an ILP system which has superseded Metagol \\cite{metagol}. Popper uses a new approach to ILP, referred to as ``Learn from failures\": while searching through the hypothesis space, it generates constraints based on incorrect programs found. Those constraints are then used to prune the search space and improve learning time. Furthermore, in \\autoref{constraints} we detail how constraints can be further used to improve the learning rate for multi-task systems.\n\n\\section{Contribution} \n\nOur contribution to the field represents an enhanced version of Popper that supports multi-task learning and uses knowledge transfer. \n\nWe have decided to implement our approach and re-implement the method described by \\citet{lin2014bias} as extensions to the Popper ILP system in order to be able to evaluate the benefit that multi-task learning can obtain by preserving constraints.\n\nSuch, we summarise our contribution as follows:\n\\begin{itemize}\n    \\item Metagol$_{DF}$ performs iterative deepening search based on the maximum size of programs in the hypothesis space. We identify situations where increasing search size is detrimental to the learning process, virtually blocking the system from progressing further. In consequence, we provide an alternative search strategy that circumvents this obstacle and has a greatly improved learning ability.\n    \\item Between each stage of iterative deepening search, the Metagol$_{DF}$ does not maintain any information. Our implementation gives Popper the ability to preserve constraints when the maximum search size is modified. Such, we improve the learning rate and allow for a breadth first search approach to replace the iterative deepening approach. For example, let $C$ be the set of constraints obtained by searching through a hypothesis space of size $n$. This constraints will help prune a search space of size $n+1$. Without constraint preservation, the system would be unable to transfer information between runs. Therefore, it must search through a space size that includes both hypotheses of size $n$ and size $n+1$.\n    \\item We show that the performance of the system is strongly correlated to the order in which the user inputs the list of tasks to be solved \\cite{ordering}. We design two methods that order tasks based on how close the system is to solving them, aiming to achieve more consistent and closer to optimal performance.\n    \\item Metagol$_{DF}$ has been tested against a limited data set of 17 string transformation operations. We refine one external data set and synthetically generate two additional ones. We then proceed to empirically evaluate the performance of all proposed techniques and indicate how each performs in comparison to the original approach.\n\\end{itemize}\n\n\\chapter{Related work}\n\n\\section{Ability to learn}\nPrograms that expand their learning abilities (``have the ability to learn\") have been an area of interest in Machine Learning for multiple decades. In \\cite{Solomonoff1989ASF}, Solomonoff describes a system that, given a small set of concepts, is able to learn problems of increasing difficulty by extracting new, more complex concepts during each learning stage. A concept was defined as a sequence of instructions. The approach reduces most problems from science to two main categories: \n\n\\textbf{Inversion problems.} Given a machine $M$, that performs string transformation operations, and a string $s$, the goal of an inversion problem is to determine a string $x$, such as $M(x) = s$. \n\n\\textbf{Time limited optimisation problems.} Given a machine $M$, that takes a string and returns a real number, the goal of a time limited optimisation problem is to find a string $x$, within a fixed time limit $T$, such as $M(x)$ is as large as possible. \n\nFurthermore, the paper concludes with an overview of areas where such a system would benefit from further research, most notably: improvements in the method used to update the set of knowledge once a new task is learnt and designing a system that is able to work on an unordered set of programs.\n\n\\section{Multi-task learning}\nIn this thesis, we refer to multiple Machine Learning paradigms related to multi-task learning. \n\n\\textbf{Meta-learning.} Meta-learning (learning to learn) focuses on how machine learning approaches perform on certain tasks and then uses information obtained from previous runs in order to speed up learning during future tasks \\cite{Joaquin2018}. \n\n\\textbf{Curriculum learning.} Curriculum learning means that tasks will be learnt in a specific order, based on their perceived difficulty (similar to a human curriculum) \\cite{XinWang2020}. \n\n\\textbf{Transfer learning.} Transfer learning is a technique through which multiple learners (that focus on related tasks) transfer knowledge to each other.  \\cite{weiss2016survey}. \n\nThe above approaches have been empirically and theoretically shown to improve performance over traditional independent learning methods \\cite{zhang2018overview}. \n\nAlthough the fact that tasks from a similar domain may be related appears intuitive, it is necessary to outline both the benefits and downsides of multi-task learning. Therefore, depending on the similarity between the given set of tasks, it is crucial to evaluate if multi-task learning is expected to provide a guaranteed advantage over independent learning \\cite{ben2003exploiting}.\nThe expectation is that multi-task learning approaches perform significantly worse than conventional learning methods on unrelated tasks \\cite{pan2010}. Moreover, transfer learning systems that retain information from all learnt tasks as part of their knowledge set face difficulty when used to solve large data sets \\cite{ferri2001incremental}. This is a consequence of the fact that search speed is usually correlated to the size of the background knowledge.\nSeveral methods that attempt to circumvent the above issues have been developed. \n\nFor example, neural networks have been used to recognise which programs are expected to perform better on specific input\/output pairs \\cite{ellis2018learning}. \n\nIt has also been experimented with the idea of ``forgetting\", which refers to removing data from the background knowledge (or ``unlearning\") with the purpose of improving search time \\cite{cropper2020forgetting}.\n\\section{Multi-task learning in ILP}\nOne of the main advantages of ILP, in the context of multi-task learning, is the fact that ILP systems can perform knowledge transfer without significant overhead through the fact that outputs are reusable as inputs. Despite such considerations, very few studies explore multi-task learning approaches for ILP. \n\nMetagol is an ILP system based on meta-interpretive learning \\cite{metagol}. Lin et al. adapt the Metagol system to make use of multi-task learning (Metagol$_{DF}$) and benchmark it against multiple agents: a human agent, a commercial system (Microsoft FlashFill, introduced with Excel 2013), and a version of itself, which does not make use of multi-task learning. \n\nThe system has shown promising results and consistently achieved better or equal performance when compared with the standard approach. \n\nThe author concludes by highlighting a series of limitations of multi-task learning in the context of ILP.\nMost importantly, it observes that the system will become overwhelmed when run on multiple data sets. This claim is empirically confirmed by Cropper, in \\cite{cropper2020forgetting}, who also provides a solution for such use cases: introducing Forgetgol. This system can forget programs and reduce the background knowledge size. \n\nIn this project, we identify significant issues with the previous approach and propose an alternative solution. We build on the Popper ILP system \\cite{Cropper2019} and introduce the following elements of novelty: we modify how the hypothesis space is searched, we retain information between failed search attempts, and we introduce new search strategies that focus on analysing the difficulty of the tasks at run-time (similar to curriculum learning).\n\n\\chapter{Problem setting}\nIn this chapter, we will give an overview of ILP in order to familiarise the reader with specific terminology used throughout the project. \n\\section{Logic programming syntax}\nWe will define the syntax of Logic Programs by following definitions from \\cite{ILPat30}, and \\cite{kowalski1974}: \n\n\\textbf{Variable.} A variable is a string of characters that starts with an uppercase letter. \n\n\\textbf{Function.} A function symbol is a string of characters that starts with a lowercase letter. Examples: \\texttt{f1}, \\texttt{move\\_up}, \\texttt{draw1}. \n\n\\textbf{Predicate symbol.} A predicate symbol is a string of characters that start with a lowercase letter, similarly to a function. \n\n\\textbf{Arity.} The arity of a function\/predicate symbol \\texttt{f} is the number of arguments it takes and is denoted as \\texttt{f\/n}. \n\n\\textbf{Constant symbol} A constant symbol is a predicate symbol of arity zero. Examples: \\texttt{alice}, \\texttt{bob}. \n\n\\textbf{Term.} A term refers to either a variable, a constant symbol or a predicate symbol of arity $n$ followed by a tuple of size $n$. \n\n\\textbf{Ground term.} We say that a term is ground if it contains no variables. \n\n\\textbf{Atom.} An atom is a formula $f(x_1,..,x_n)$, such as $p$ is a predicate symbol of arity $n$ and $x_i$ is a term $\\forall i \\in \\{1,..,n\\}$. \n\n\\textbf{Ground atom.} We say that an atom is ground if all its terms are ground. \n\n\\textbf{Negation.} We say that an atom is false if it cannot be proven true. Throughout this paper, we will represent negation through the symbol $\\neg$. \n\n\\textbf{Literal.} A literal is either an atom or its negation. \n\n\\textbf{Clause.} A clause is a disjunction of atoms $h_1,h_2,..,h_n,b_1,b_2,..,b_m$, represented as:\n\\begin{align*}\n    h_1,h_2,..,h_n \\textrm{ :- } b_1,b_2,..,b_m\n\\end{align*}\nThe above definition is equivalent to:\n\\begin{align*}\n    (h_1\\land h_2\\land..\\land h_n) \\lor \\neg b_1 \\lor \\neg b_2 .. \\lor \\neg b_m\n\\end{align*}\n\nWe refer to $h_1,h_2,..,h_n$ as the \\textit{head literals} and  $b_1,b_2,..,b_m$ as the \\textit{body literals} of the clause. \n\nThe above representation is shorthand for ``$h$ holds true if all of $b_1,..,b_m$\" hold true. \n\n\\textbf{Horn clause} A Horn clause is a clause for which $n$ (number of head\/positive literals) is at most $1$. \n\nWe will only be concerned with Horn clauses since other types of clauses are not supported by basic Prolog \\cite{kowalski1988early}. \n\n\\textbf{Ground clause.} We say that a clause is ground if it contains no variables. \n\n\\textbf{Definite clause.} We say that a Horn clause is definite if the number of positive literals is exactly one. \n\n\\textbf{Definite logic program.} A definite logic program is a set of definite clauses. We will further refer to definite logic programs as programs.\n\\section{Inductive logic programming}\nInductive logic programming problems are given as triplets $(B,E^+,E^-)$, where $B$ represents the background knowledge (BK), $E^+$ represents the set of positive examples, and $E^-$ represents the set of negative examples. \n\n\\textbf{Background knowledge.} The background knowledge is a set of programs for which the definition is given completely. By the close world assumption, anything that is not explicitly stated true in the BK is considered false \\cite{REITER1981119}. We show a specimen from the BK of the string transformation problem: \n\n\\[\n\\texttt{B} = \\left\\{\n\\begin{array}{l}\n    \\texttt{is\\char`_empty([]).}\\\\\n    \\texttt{is\\char`_space([' '|\\char`_]).}\\\\\n    \\texttt{is\\char`_number(['0'|\\char`_]).}\\\\\n    \\texttt{...} \\\\\n    \\texttt{is\\char`_number(['9'|\\char`_]).}\\\\\n\\end{array}\n\\right\\}\n\\]\n\n\\textbf{Examples.} Examples are given as two sets (positive and negative examples) and are represented similarly to the BK, but all clauses must be ground. Moreover, there is no assumption on the function behaviour outside of the given use cases. Below we show examples of a function that turns the first letter of a string to uppercase. \n\n\\[\n\\texttt{E$^+$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['j', 'o', 'z', 'i', 'e'],['J', 'o', 'z', 'i', 'e']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B', 'e', 'n']).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['j', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['j', 'o', 'z', 'i', 'e'],['o', 'z', 'i', 'e']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B']).}\\\\\n\\end{array}\n\\right\\}\n\\]\n\n\\textbf{Hypothesis space.} The hypothesis space contains all the possible programs that can be built from the predicates given in the BK. Since the hypothesis space is infinite, it is important to enforce certain restrictions to make searching through it practical. Such, many machine learning makes use of inductive bias and language bias \\cite[p,64,106]{Mitchell97} to restrict the hypothesis space. \n\\section{Popper ILP System}\nIn \\autoref{implementation} we introduce our multi-task learning strategies, which build on the Popper ILP system. The Popper ILP system uses an approach known as ``Learning from failures\" \\cite{popper}. This technique divides the learning process into three stages: \\textbf{generate}, \\textbf{test}, and \\textbf{constrain}. \n\nIn the generate phase, a logic program (referred to as hypothesis) is selected, such that it respects the currently learnt \\textbf{hypothesis constraints}. Afterwards, the program is tested against the given examples. If all examples are covered, Popper will return the program as a valid answer. Otherwise, we proceed to the constrain stage, where information from the testing stage is used in order to augment the set of \\textbf{hypothesis constraints}. Those three stages are repeated until either a solution is found or there are no more programs in the hypothesis space. \n\nBelow we will describe the four types of constraints. \n\n\\textbf{Generalisation constraints.} A generalisation constraint is a constraint that eliminates generalisations of the current hypothesis. Example: \n\nGiven the following negative examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B', 'e', 'n']).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that a negative example holds for $h$. Such, we can prune generalisations, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\texttt{f(A, B):-head(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Specialisation constraints.} A specialisation constraint is a constraint that prunes out specialisations of the current hypothesis. Example: \n\nGiven the following positive examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).} \\\\\n\\texttt{f(['b', 'e', 'n',' ', 'g'],['B', 'e', 'n',' ', 'G']).} \\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that the first positive example holds for $h$, but not the second. Such, we conclude that $h$ is too specific and prune out specialisations of $h$, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B),empty(B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Elimination constraints.} An elimination constraint is a constraint that eliminates a hypothesis $h$ from any separable hypothesis $h_s$ that contains $h$. We refer to a hypothesis $h_s$ as separable if no predicate symbol from the head of a clause in $h_s$ appears in the body of a clause in $h_s$. Example: \n\nGiven the following positive examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).} \\\\\n\\texttt{f(['b', 'e', 'n',' ', 'g'],['B', 'e', 'n',' ', 'G']).} \\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-empty(A).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that the no positive example holds for $h$. Such, we conclude that $h$ can be eliminated from any separable hypothesis, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B)}\\\\\n\\texttt{f(A, B):-empty(A).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Banish constraints.} A banish constraint is a constraint that rules out one specific hypothesis. \n\nTo handle searching through a vast hypothesis space size, the Popper ILP system looks for solutions in increasing order of the number of literals in a hypothesis. The formula used to compute an upper bound on the size of the hypothesis space is given in \\cite{popper}: \n\\begin{itemize}\n    \\item Maximum arity $\\boldsymbol{a}$.\n    \\item Maximum number of unique variables in a clause $\\boldsymbol{v}$.\n    \\item Maximum number of body literals allowed in a clause $\\boldsymbol{m}$.\n    \\item Maximum number of clauses in a hypothesis $\\boldsymbol{n}$.\n    \\item Declaration bias $\\boldsymbol{D}=(D_h,D_b)$.\n\\end{itemize}\n\n\\begin{align*}\n    \\sum_{j=1}^{n}\\binom{|D_h| v^a \\sum_{i=1}^{m}\\binom{|D_b| v^a}{i}}{j}\n\\end{align*}\n\\label{ILPintro}\n\\chapter{Implementation}\n\\label{implementation}\nIn this section, we will describe several multi-task learning approaches. For each one, we will detail the advantages and disadvantages. We begin with the naive approach, followed by the current state-of-the-art strategy. Afterwards, we will provide a new search algorithm that focuses on resetting the size of the search space in order to overcome the limitations of the state-of-the-art approach. Finally, we will describe two refinements that can be applied to any of the described approaches: constraint preservation and automatic task ordering.\n\n\\section{Naive approach}\n\\noindent This naive approach to multi-task learning iterates through all the tasks in order and attempts to learn them by calling an ILP system (in our case Popper) \\footnote{We note that this search strategy can get blocked when working on unsolvable tasks. This behaviour does not impact our results as the naive approach is only used for reference and we will benchmark against the state-of-the-art strategy (which is discussed in \\autoref{sota-sub}). Other approaches that do not make use of multi-task learning can be designed to overcome this difficulty.}. Therefore, it does not make use of the fact that tasks learned together are expected to be correlated. \n\n\\paragraph{Advantages.} The main advantage of this approach is that it does not suffer from the additional overhead introduced by increasing the number of tasks in the BK \\cite{lin2014bias}. In \\autoref{fig:naive-example} the system is given two independent tasks: learning the \\texttt{isGrandfather} and \\texttt{isGrandmother} functions. In such a situation, transferring knowledge between the two will be redundant and only cause unnecessary overhead. Therefore, such use-cases benefit the naive approach.\n\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n    \n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isMother(\"Alice\",\"Dan\")\n        isMother(\"Ana\", \"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Ana\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isMother(A,C), isMother(C, B)\n        \n    \\end{minted}\n    \\caption{Advantage example for naive approach}\n    \\label{fig:naive-example}\n\\end{figure}\n\n\\paragraph{Disadvantages.} Although it performs well on uncorrelated data, multi-task learning systems have been shown repeatedly to out-perform traditional systems on tasks from the same domain \\cite{ILPat30}. In \\autoref{fig:bad-naive-example} the system is given the same two tasks, but the BK is modified such that the tasks are no longer independent: \n\n\\texttt{isGrandmother} can be learnt as a function of \\texttt{isGrandfather}. The naive approach has no provision to benefit from such situations.\n\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n    \n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)\n        \n    \\end{minted}\n    \\caption{Disadvantage example for naive approach}\n    \\label{fig:bad-naive-example}\n\\end{figure}\n\nFinally, in \\autoref{alg:multi_popper_naive} we show an implementation of the naive multi-task learning approach.\n\n\\begin{algorithm}[hbt!]\n\\caption{Naive approach to multi-task learning through Popper}\\label{alg:multi_popper_naive}\n\\begin{algorithmic}\n\\Function{multipoppernaive}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\ForAll{$t$ in $T$}\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $BK$, $\\mathit{size}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\State $\\mathit{size} \\gets \\mathit{size}+1$ \n\\EndWhile\n\\EndFor\n\\State \\Return $S$\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Knowledge transfer: iterative deepening approach}\n\\label{sota-sub}\nAs previously discussed, the naive approach does not make use of knowledge transfer. To address this issue, in \\cite{lin2014bias}, Lin et al. propose a new multi-task learning approach based on the Metagol ILP system. This method is currently state-of-the-art in multi-task ILP. The described solution uses iterative deepening search, proceeding as follows:\n\\begin{enumerate}\n    \\item Set $\\mathit{maxSize} = 1$.\n    \\item For all tasks, try to find a solution with size at most $\\mathit{maxSize}$.\n    \\item If all tasks are solved, return.\n    \\item Otherwise, update the background knowledge with newly learnt tasks, increment $\\mathit{maxSize}$ by 1, and return at step 2.\n\\end{enumerate}\n\nPseudo-code for integrating this search strategy into the Popper ILP system is present in \\autoref{alg:multi_popper_soda}. \n\n\\begin{algorithm}[hbt!]\n\\caption{Iterative deepening search}\\label{alg:multi_popper_soda}\n\\begin{algorithmic}\n\\Function{multipopper}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{maxSize} \\gets 1$\n\\While{TRUE}\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $\\mathit{BK}$, $\\mathit{maxSize}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\EndIf\n\\EndFor\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\State $\\mathit{maxSize} \\gets \\mathit{maxSize} + 1$\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\paragraph{Advantages.} Through the introduction of knowledge transfer as part of the multi-task learning strategy, this solution improves performance over the naive approach when used on correlated data sets. For example, assume our background knowledge contains two primitive predicates \\texttt{bk0} and \\texttt{bk1} and we are given two tasks, \\texttt{f0} and \\texttt{f1}, such that \\linebreak \\texttt{f1(A,B,C):-f0(A),f0(B),f0(C)}. Learning \\texttt{f0} and re-using it's definition as part of an enhanced background knowledge allows us to learn \\texttt{f1} more quickly by reducing the size of its solution.\n\\paragraph{Disadvantages.} We observe that even when the BK is augmented, the search size continues to increase. On one hand, this particularity provides an advantage when solution sizes remain large even with the new BK, such as in \\autoref{ref:soda-example}. On the other hand, it gives rise to the following disadvantages: \n\n\\begin{figure}[hbt!]\n\\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        isMother(\"Mary\",\"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandgrandmother(\"Mary\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandgrandmother(A,B) :- isMother(A,C), isWife(C,D),\n                                   isGrandfather(D, B)\n\\end{minted}\n\\caption{Positive example for iterative deepening search}\n\\label{ref:soda-example}\n\\end{figure}\n\nFirstly, after each increase of the $\\mathit{maxSize}$ variable, we will reevaluate all programs of size between $1$ and  $\\mathit{maxSize} - 1$ from the previous iteration. \n\nSecondly, we might search through an unnecessarily large hypothesis space when a BK augmentation might have allowed us to find a solution in a smaller space. \n\nMoreover, as a consequence of the second disadvantage, this approach does not learn the most compact solution possible. We will further describe this case below:\n\\begin{enumerate}\n    \\item We will use the BK from the robot movement problem. For simplicity, we will only be concerned with \\texttt{move\\_up(A,B)}.\n    \\item Assume we have two tasks, \\texttt{f0(A,B)} and \\texttt{f1(A,B)}, such that their solutions are:\n    \\begin{itemize}\n        \\item \\texttt{f0(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,B)}\n        \\item \\texttt{f1(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,E),move\\_up(E,B)}\n    \\end{itemize}\n    \\item In this approach, we will find the solution for \\texttt{f0} after having $\\mathit{maxSize}=4$, and then increment $\\mathit{maxSize}$ to 5, before augmenting the BK.\n    \\item When searching for programs up to size 5, we can identify two valid solutions for \\texttt{f1}:\n    \\begin{itemize}\n        \\item  \\texttt{f1(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,E),move\\_up(E,B)}\n        \\item  \\texttt{f1(A,B):-f0(A,C),move\\_up(C,B)}\n    \\end{itemize}\n    \\item Such, we observe that the algorithm could return either solution, having no preference for the more compact program.\n\\end{enumerate}\n\nThirdly, this approach may lead to situations where learning does not conclude in a reasonable time frame because of the increase in the hypothesis space size. This can be encountered in a situation where functions a learnt in a chain, for example:\n\\begin{enumerate}\n    \\item Define $\\texttt{p(A,B)}^n \\texttt{:-} \\texttt{p(A,C),p(C,B)}^{n-1}$ for any predicate \\texttt{p} and natural number $n > 1$. Base case $\\texttt{p(A,B)}^1 \\texttt{:-} \\texttt{p(A,B)}$. Then, we have tasks \\texttt{f0,f1,f2,f3,f4}:\n    \\begin{itemize}\n        \\item $\\texttt{f0(A,B):-move\\_up(A,C)}^6$\n        \\item $\\texttt{f1(A,B):-move\\_up(A,C)}^{36}$\n        \\item $\\texttt{f2(A,B):-move\\_up(A,C)}^{216}$\n        \\item $\\texttt{f3(A,B):-move\\_up(A,C)}^{1296}$\n        \\item $\\texttt{f4(A,B):-move\\_up(A,C)}^{7776}$\n    \\end{itemize}\n    \\item By using this approach, \\texttt{f0} has to be learnt and added to the BK before \\texttt{f1}, \\texttt{f1} before \\texttt{f2}, \\texttt{f2} before \\texttt{f3}, \\texttt{f3} before \\texttt{f4}.\n    \\item Such, each task will be learnt by searching through hypothesis spaces of sizes as follows:\n    \\begin{itemize}\n        \\item \\texttt{f0} at size 7\n        \\item \\texttt{f1} at size 8\n        \\item \\texttt{f2} at size 9\n        \\item \\texttt{f3} at size 10\n        \\item \\texttt{f4} at size 11\n    \\end{itemize}\n    \\item But, all given tasks will have solutions of 6 body literals:\n    \\begin{itemize}\n        \\item $\\texttt{f0(A,B):-move\\_up(A,C)}^6$\n        \\item $\\texttt{f1(A,B):-f0(A,C)}^{6}$\n        \\item $\\texttt{f2(A,B):-f1(A,C)}^{6}$\n        \\item $\\texttt{f3(A,B):-f2(A,C)}^{6}$\n        \\item $\\texttt{f4(A,B):-f3(A,C)}^{6}$\n    \\end{itemize}\n    \\item Given the formula from \\autoref{ILPintro} and previous experiments we know that hypothesis spaces containing programs of more than 10 literals can be generally considered unsolvable within a reasonable time limit. Such, we observe the benefit of the new approach.\n\\end{enumerate}\n\nIn the rest of this chapter, we will propose and analyze multiple search strategies and improvements to overcome the aforementioned disadvantages.\n\n\\subsection{Knowledge transfer: resetting search size}\nWe proceed by bringing forward a new search strategy and providing an argument for its theoretical advantages over the current state-of-the-art approach.\n\nWhen concerned with programs that achieve a significant reduction in size through knowledge transfer, the previous approach performs redundant work by continuing to search for solutions of constantly increasing maximum size. We argue for restarting the search from maximum size to one after a solution is found to one of the tasks.\n\nBy doing this we reduce the amount of work required when programs become ``easy\" after knowledge transfer is performed. Below we will compute the difference in the number of candidate solutions between the two approaches:\n\\begin{enumerate}\n    \\item Let $\\mathit{BK}_0$ be our initial background knowledge.\n    \\item Let $\\mathit{BK}_1 = \\mathit{BK}_0 \\cup \\{\\texttt{f0}\\}$.\n    \\item Let \\texttt{f0}, \\texttt{f1} be two predicates such that:\n    \\begin{itemize}\n        \\item \\texttt{f0} has a solution of size $n$ in $\\mathit{BK}_0$.\n        \\item \\texttt{f1} has a solution of size $m > n$ in $\\mathit{BK}_0$.\n        \\item \\texttt{f1} has a solution of size $p < m$ in $\\mathit{BK}_1$.\n    \\end{itemize}\n    \\item Let $\\mathit{HS}(\\texttt{f}, \\mathit{size}, \\mathit{BK})$ be the size of the hypothesis space for task \\texttt{f}, given $\\mathit{size}$ and the background knowledge $\\mathit{BK}$.\n    \\item Let $\\mathit{HS}^*(\\texttt{f}, \\mathit{maxSize}, \\mathit{BK}) = \\sum\\limits_{\\mathit{size}=1}^{\\mathit{maxSize}}\\mathit{HS}(\\texttt{f}, \\mathit{size}, \\mathit{BK})$\n    \\item Then, the total number of programs generated can be computed as follows:\n    \\begin{itemize}\n        \\item Without reset we have: \\begin{dmath}\\sum\\limits_{i=1}^{n}(\\mathit{HS}^*(\\texttt{f0}, i, \\mathit{BK}_0) + \\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_0)) +\\\\ \\mathit{max}(\\sum\\limits_{i=n + 1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1),\\mathit{HS}^*(\\texttt{f1}, n + 1, \\mathit{BK}_1))\\end{dmath}\n        \\item With reset we have: \\begin{dmath}\\sum\\limits_{i=1}^{n}(\\mathit{HS}^*(\\texttt{f0}, i, \\mathit{BK}_0) + \\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_0)) + \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n        \\end{dmath}\n    \\end{itemize}\n    \\item Therefore, we identify two cases:\n    \\begin{itemize}\n        \\item If $p \\leq n$ (4.1) and (4.2) can be written and compared as follows:\n                \\begin{dmath}\n                    \\mathit{HS}^*(\\texttt{f1}, n + 1, \\mathit{BK}_1) > \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n                \\end{dmath}\n        \\item If $p > n$ (4.1) and (4.2) can be written and compared as follows:\n            \\begin{dmath}\n                \\sum\\limits_{i=n + 1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1) < \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n            \\end{dmath}\n    \\end{itemize}\n    \\item By using the formula for $\\mathit{HS}$, given in \\autoref{ILPintro}, we argue that the performance increase in the case where $p \\leq n$ overcomes the performance decrease from the case $p > n$. Moreover, we can extend this argument to multiple tasks (e.g. introduce \\texttt{f2} such that \\texttt{f2} is very difficult, but has a much simpler solution that uses \\texttt{f1}) in order to strengthen the argument.\n\\end{enumerate}\n\nAs a result, we can observe one use case improvement of the reset strategy. \n\nA more concrete example can be seen in \\autoref{fig:reset-example}, where the original approach would have performed redundant work on learning the isGrandgrandmother task by evaluating solutions of size four when a solution can be found at size three. \n\n\\begin{figure}[hbt!]\n\\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        isMother(\"Olga\",\"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n        isGrandgrandmother(\"Olga\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C, B)\n        \n        isGrandgrandmother(A,B) :- isMother(A,C), isGrandmother(C,B)\n\\end{minted}\n\\caption{Example for iterative deepening search with search size reset}\n\\label{fig:reset-example}\n\\end{figure}\n\nMoreover, we will formalise this claim:\n\\begin{enumerate}\n    \\item Assume there exists some task $f$, such that a solution of size $n$ has been found, but there exists a simpler solution, of size $m$, $m < n$. Both solutions generated from the same BK.\n    \\item By our approach, if a solution of size $n$ has been found then Popper tried all solutions of size $s$, $s\\in \\{1,2,..,n-1\\}$ with the current BK.\n    \\item Then it means that no solution exists of size $s$, $s \\in \\{1,2,..,n-1\\}$.\n    \\item Contradiction with $m < n$.\n    \\item Therefore, $n$ is the smallest solution size that can be found with the current BK.\n\\end{enumerate}\n\nAn \\autoref{alg:multi_popper_reset} we showcase pseudo-code for the reset approach.\n\n\\begin{algorithm}[hbt!]\n\\caption{Iterative deepening search with search size reset}\\label{alg:multi_popper_reset}\n\\begin{algorithmic}\n\\Function{multipopperreset}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{maxSize} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $\\mathit{BK}$, $\\mathit{maxSize}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{maxSize} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{maxSize} \\gets \\mathit{maxSize} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Knowledge transfer: removing redundant testing}\nRecall \\autoref{alg:multi_popper_reset}. Any call to Popper would search all solutions up to $\\mathit{maxSize}$, each time it is incremented. We can modify the Popper algorithm to only search for programs of a fixed size. Therefore, we will no longer evaluate programs twice unless a solution has been found to one of the tasks. An overview of this change is present in \\autoref{alg:multi_popper_reset+}. \n\\paragraph{Advantages.} The main advantage of this approach is that it does not reiterate through smaller search space sizes unless necessary (when the BK is augmented).\n\\paragraph{Disadvantages.} Because we do not re-search the smaller hypothesis space sizes, we do not keep constraint information between searches. This may result in a significant increase in the size of the hypothesis space evaluated. More specifically, if we are searching through a hypothesis space that contains only programs of size $n$, we will not make use of the constraint generated in the previous steps from programs of sizes between $1$ and $n - 1$.\n\\begin{algorithm}[hbt!]\n\\caption{Breadth fist search with size reset}\\label{alg:multi_popper_reset+}\n\\begin{algorithmic}\n\\Function{multipopperreset+}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{fixedsizedpopper}{$t$, $\\mathit{BK}$, $\\mathit{size}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{size} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{size} \\gets \\mathit{size} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\section{Preserving constraints}\n\\label{constraints}\nTransferring knowledge between tasks requires Popper to retry hypothesis spaces that contain programs with a smaller number of literals than the maximum tried until now (because the introduction of more programs in the background knowledge creates new possible solutions at the respective sizes). Therefore, all presented approaches will retry the same set of solutions multiple times. \n\nWhen modifying search size, the Metagol did not have the ability to transfer knowledge through a standard form, even when working with a single task, e.g. if you have not found a solution of maximum size up to $n$, you would need to restart the search from scratch when looking for a solution of maximum size up to $n + 1$. In this project, we circumvent this limitation by extending the Popper ILP system, which has the ability to retain information from failed attempts in the form of hypothesis constraints.\n\nMore specifically, Popper allows us to retain constraints between failed attempts and prune the hypothesis space as soon as we begin searching with different parameters (in our case, an increased maximum number of literals).\n\nTo reduce the amount of repeated work caused by search size resets, we adapt all previous methods to preserve constraints learnt by Popper for each task. \n\nWe showcase that necessary changes to implement constraint preservation in the improved reset approach (recall \\autoref{alg:multi_popper_reset+}) in \\autoref{alg:multi_popper_cons}. \n\\begin{algorithm}[hbt!]\n\\caption{Constraint preserving search}\\label{alg:multi_popper_cons}\n\\begin{algorithmic}\n\\Function{multipoppercons}{}\n\\For{$i$ in \\Call{range}{n}}\n    \\State $\\mathit{rules}[i] \\gets \\emptyset$\n\\EndFor\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S \\gets \\emptyset$\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $(\\mathit{sol},\\mathit{cons}) \\gets$ \\Call{constraintpopper}{$t$, $\\mathit{BK}$, $\\mathit{size}$, $\\mathit{rules}[i]$}\n    \\State $\\mathit{rules}[i] \\gets \\mathit{rules}[i] \\cup \\mathit{cons}$\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{size} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{size} \\gets \\mathit{size} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\paragraph{Advantages.} By making use of constraint preservation, we remove the bottleneck created by resetting the search on a program and then retesting all candidate hypotheses. Moreover, this technique resolves the significant disadvantage of the breadth first search approach. Now that all constraints are preserved, there is no information lost by not reiterating through the smaller hypothesis spaces.\n\\paragraph{Disadvantages.} We argue that there do not exist significant disadvantages of preserving constraints. One could point out the fact that they may increase memory requirements and the overhead of the reset phase. However, the expectation is that unless a solution is found early, most of the constraints from previous stages would have also been learnt during the current iteration. \n\nFinally, we provide an argument for the fact that it is always correct to preserve constraints. \n\nWe are given the fact that any constraint generated by the system is sound \\cite{popper}. \n\nGiven a constraint $C$ learnt during an iteration of the learning algorithm with background knowledge $\\mathit{BK}$, we show that constraint $C$ would also hold for background knowledge $\\mathit{BK}' = \\mathit{BK} \\cup N$, where $N$ is some set of valid background knowledge. \n\nLet $h$ be the hypothesis that determined $C$. $h$ is generated from $\\mathit{BK}$, therefore  $h$ also generated from $\\mathit{BK'}$. Therefore, during a learning process based on $\\mathit{BK'}$, if $h$ will also be tested, it will generate the same sound constraint. \n\\section{Automatic computation of task learning order}\nWhen considering large data sets where some tasks may be unsolvable, the ordering of the tasks becomes important because the algorithm may not get to consider certain tasks that have a solution of the current search size. Consequently, we propose the introduction of task ordering methods, similar to curriculum learning. We focus on run-time evaluation of tasks, aiming to order tasks by how close our system is to finding a solution. For example, take the following scenario\\footnote{For this example we assume that the solution program is always found last in the hypothesis space}:\n\\begin{enumerate}\n    \\item We are given 100 tasks, \\texttt{fi} with \\texttt{i} $\\in \\{0,..,99\\}$.\n    \\item Task \\texttt{f99} has a solution of size seven.\n    \\item All other tasks have solutions of nine using the initial BK and solutions of size three if \\texttt{f99} is part of their BK.\n    \\item Recall the formula in \\autoref{ILPintro}. Based on it, we will approximate the number of programs in hypothesis space of size $x$ to $2^{2^x}$ \\footnote{The purpose of this approximation is to provide the reader with an overview of the rate at which the size of the hypothesis space increases with the number of literals. Providing a more accurate result would require a discussion around the values of other parameters from the formula in \\autoref{ILPintro}.}. \n    \\item Now we will compare the solving time for the reset approach of the best case and worst case orderings.\n    \\begin{itemize}\n        \\item Order \\texttt{f0}, \\texttt{f1},..,\\texttt{f99}\n        \\begin{dmath}\n            \\sum\\limits_{i=1}^{7}(100*2^{2^i})+\\sum\\limits_{i=1}^{3}(99*2^{2^i})\n        \\end{dmath}\n        \\item Order \\texttt{f99}, \\texttt{f0},\\texttt{f1},\\texttt{f2},..,\\texttt{f98} (\\texttt{f99} is computed first)\n        \\begin{dmath}\n            \\sum\\limits_{i=1}^{6}(100*2^{2^i})+2^{2^7}+\\sum\\limits_{i=1}^{3}(99*2^{2^i})\n        \\end{dmath}\n    \\end{itemize}\n    \\item We obtain that when the system is given the order \\texttt{f0}, \\texttt{f1},..,\\texttt{f99} it tests approximately 100 times more programs then when it is given the order \\texttt{f99}, \\texttt{f0},\\texttt{f1},\\texttt{f2},..,\\texttt{f98}.\n\\end{enumerate}\n\\paragraph{Advantages.} Given the above example, we know that task ordering is important for the performance of the system. Approaches that perform automatic task ordering can overcome this limitation, by identifying such scenarios and selecting and appropriate task sequence.\n\n\\paragraph{Possible disadvantages.} We outline a list of possible disadvantages in order to highlight the points we need to take into account when designing our heuristics. Firstly, the evaluation function is not guaranteed to provide a significant benefit. For example, a heuristic that orders by the number of examples entailed by the best program yet would have no effect on a data set where all tasks are given a single example. Secondly, wrongly estimating the importance of size in the evaluation function may significantly deter search time by testing more extensive programs when not necessary. To mitigate this issue, we experiment with a conservative evaluation function: firstly, ordering by size and then tie-breaking through our heuristic. \n\nGiven the above considerations, we propose two solutions for this problem: one that orders tasks by the number of examples currently solved and the other by the number of constraints generated by the system.\n\\subsection{Ordering by number of examples solved}\nThe first approach we experiment with is ordering tasks decreasingly by the number of examples currently solved. We motivate this finding on the fact that very often, entailing an example is equivalent to finding a program that has at least one clause which is part of a solution.\n\\subsection{Ordering by number of constraints}\nThe second approach we experiment with is ordering tasks by the number of constraints the system has generated for them. If the system has generated more constraints for a program it means that its hypothesis space can be pruned the most. Therefore, we expect this approach to identify tasks that are closed to being solved.\n\nIn \\autoref{alg:multi_popper_prio} we provide a generic implementation of the task order approach. It makes use of a priority queue data structure \\textit{TPQ}, which is initialised with the desired heuristic.\n\n\\begin{algorithm}[hbt!]\n\\caption{Priority searching of solutions}\n\\label{alg:multi_popper_prio}\n\\begin{algorithmic}\n\\Function{multipopperprio}{}\n\\For{$i$ in \\Call{range}{n}}\n    \\State $\\mathit{rules}[i] \\gets \\emptyset$\n\\EndFor\n\\State $TPQ \\gets \\{(t_1, 1, 1.0), (t_2, 1, 1.0),..,(t_n, 1, 1.0)\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S \\gets \\{\\}$\n\\While{\\textbf{not} TPQ \\textbf{is empty}}\n\\State $(t, size, \\_)$ $\\gets$ \\Call{pop}{$TPQ$}\n\\State $(\\mathit{partialsol},\\mathit{cons}) \\gets$ \\Call{popperpartial}{$t$, $BK$, $\\mathit{size}$, $\\mathit{rules}[i]$}\n\\State $\\mathit{rules}[i] \\gets \\mathit{rules}[i] \\cup \\mathit{cons}$\n\\If{\\Call{isvalidsol}{$\\mathit{partialsol}$,$t$}}\n\\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\State \\Call{ResetPopper}{T}\n\\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n\\State $TPQ \\gets \\{(t_i, 1, 1.0) \\vert t_i \\in T \\}$\n\\Else\n\\If{$partialsol \\neq null$}\n\\State $priority \\gets$ \\Call{computeprio}{$sol$}\n\\State \\Call{PUSH}{$TPQ$, $(t, size + 1, priority)$}\n\\EndIf\n\\EndIf\n\\EndWhile\n\\State \\Return $S$\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\chapter{Experimental Results}\nThe goal of this chapter is to determine the performance of the approaches proposed in the previous chapter. We will perform experiments in order to obtain answers to the following questions: \n\n\\textbf{Q1.} Does constraint preservation offer an improvement in learning speed for approaches that repeat searching over the same hypothesis space? \n\nFor each strategy mentioned previously, we will evaluate the running time change resulting from enabling constraint preservation on multiple data sets that present a wide array of specific features. \n\n\\textbf{Q2.} Do the advantages of resetting search size overcome its disadvantages when compared to the state-of-the-art approach? \n\nWe will compare the two reset strategies with the current state-of-the-art iterative deepening search approach on data sets that are expected to benefit each method and compare the best case and worst case results for each algorithm. \n\n\\textbf{Q3.} Is there any benefit in heuristically ordering programs on the same size level during a search? \n\nWe will compare the two ordering strategies against a system that selects a random ordering at the beginning of the learning process and respects it throughout each stage. We have chosen this benchmark in order to remove any bias from a system that takes the given original order, which may be beneficial\/not beneficial. \n\nThe scope of this experiment will be to determine if heuristic program ordering is either beneficial, neutral, or provides a drawback through the added overhead.\n\\section{Data sets}\n\\textbf{Strings transformation.} We test our system using an existing data set of string transformation operations, out of which we select the ones that have been found solvable by Popper during previous experiments \\cite{string-data}.\n\n\\textbf{Robot movement.} We generate a set of tasks that focus on robot movement. The goal of each task is to learn to traverse a large area using only unit length movement instructions. This task will be used to compare the performance of the reset strategy and state-of-the-art strategy on a large set of tasks that can all be reduced to a size of three literals in a multi-task learning setting. \n\n\\textbf{Printer head.} We generate a set of tasks that simulate the movement of a printer head over a piece of paper. The printer head is able to perform one pixel size movements, fill a pixel with colour or return to an initial position. \n\nThis data set is an extension of the robot movement set, which also allows recursion. Instead of learning to traverse fixed distances, the printer head learns to work on sheets of arbitrary size. \n\nIn \\autoref{fig:zebra} and \\autoref{fig:cross} we show two examples of pixel images used in training the system.\n\\begin{figure}[!htb]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.4\\linewidth]{images\/zebra.png}\n        \\caption{Zebra pattern image}\n        \\label{fig:zebra}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.4\\linewidth]{images\/cross.png}\n    \\caption{Cross pattern image}\n    \\label{fig:cross}\n    \\end{minipage}\n\\end{figure}\n\\section{Testing environment}\n\\subsection{Specifications}\nAll experiments have been run on Amazon EC2 machines (c6i.large), which have the following specifications: \n\n\\textbf{Operating system:} Ubuntu 20.04.3 LTS\n\n\\textbf{Software versions:}\n\\begin{itemize}\n    \\item \\textbf{Python} 3.8.10\n    \\item \\textbf{PySwip} 0.2.11\n    \\item \\textbf{Clingo} 5.5.0\n    \\item \\textbf{SWI-Prolog} 8.4.2\n\\end{itemize}\n\\textbf{Processor:} Intel(R) Xeon(R) Platinum 8375C CPU @ 2.90GHz \n\n\\textbf{Number of cores:} 2 \n\n\\textbf{Threads per cores:} 2 \n\n\\textbf{RAM memory:} 4 GB\n\\subsection{Testing protocol}\nFor all data sets, each strategy was run ten times with a timeout limit of 6000 seconds. \n\nBefore each run, the ordering of the input tasks is randomly determined. This approach allows our results to ensure the reliability of our data and provide greater confidence that the observed performance of the system is consistent. \n\nThere are significant sources of noise that require experiments to be run multiple times. For example, some input orderings benefit some systems over others. Furthermore, the Popper system searches through a hypothesis space in a non-deterministic manner. Therefore, during different runs, specific tasks can be given different definitions that are not necessarily reusable in the same contexts.\n\\section{Results}\n\\subsection{Algorithms}\nIn this section, we will define a list of acronyms for each of our search strategies in order to make it easier for the reader to interpret the results. \n\n\\textbf{NAIVE} is the naive algorithm. \n\n\\textbf{ID} is the state-of-the-art, iterative deepening approach. \n\n\\textbf{RESET-ID} is the iterative deepening reset approach. \n\n\\textbf{RESET-BFS} is the breadth first search reset approach. \n\n\\textbf{PRIO-EX} is a version of RESET-BFS that sorts tasks on the same level decreasingly by the number of examples solved. \n\n\\textbf{PRIO-CONS} is a version of RESET-BFS that sorts tasks on the same level decreasingly by the number of constraints discovered. \n\nIf any algorithm name is followed by the PRESERVE keyword (e.g. ID PRESERVE), it means that constraint preservation option has been set true. Otherwise, we assume that constraint preservation has not been used. \n\nBecause the breadth first search based algorithms perform poorly without constraint preservation, we will choose always to enable constraint preservation when comparing the performance of multiple strategies.\n\\subsection{Printer head}\n\\subsubsection{Overview}\nIn \\autoref{fig:printer-results} we showcase the experimental results for all our algorithms as a percentage of tasks solved after several minutes. This data set contains many tasks that are solved relatively early and a few complicated tasks, such we attach an additional plot (see \\autoref{fig:printer-sec-results}) that highlights the behaviour of the strategies in the first 100 seconds. We consider that this metric better describes the performance of the systems because it more clearly showcases their differences and removes the redundant noise generated by task ordering in the higher hypothesis spaces.\n\\pgfplotsset{\n    myplotstyle\/.style={\n    legend style={draw=none, font=\\small},\n    legend cell align=left,\n    legend pos=outer north east,\n    ylabel style={align=center, font=\\bfseries\\boldmath},\n    xlabel style={align=center, font=\\bfseries\\boldmath},\n    x tick label style={font=\\bfseries\\boldmath},\n    y tick label style={font=\\bfseries\\boldmath},\n    ymax = 100,\n    scaled ticks=false,\n    every axis plot\/.append style={thick},\n    },\n}\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 80,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of printer head results (showing standard error)}\n    \\label{fig:printer-results}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Zoom in of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:printer-sec-results}\n\\end{figure}\n\n \\begin{table}[hbt!]\n \\centering\n\\begin{tabular}{c|c|c|c}\n        Algorithm & \\multicolumn{1}{l|}{Min} & \\multicolumn{1}{l|}{Max} & \\multicolumn{1}{l}{Standard deviation} \\\\ \\hline\nNAIVE              & 1                        & 9                       & 1.97                                   \\\\\nPRIO-CONS          & 7                       & 10                       & 1.1                                   \\\\\nID               & 11                       & 13                       & 0.7                                   \\\\\nID PRESERVE      & 11                       & 13                       & 0.65                                    \\\\\nRESET-ID           & 11                       & 13                       & 0.6                                   \\\\\nRESET-ID PRESERVE  & 11                        & 13                       & 0.62                                  \\\\\nRESET-BFS PRESERVE & 11                       & 13                       & 0.64                                  \\\\\nPRIO-EX            & 11                       & 13                       & 0.66                                   \\\\\nPRIO-EX PRESERVE   & 11                       & 13                       & 0.63                                    \\\\\nPRIO-CONS PRESERVE & 12                       & 13                       & 0.46                                  \\\\\nRESET-BFS          & 11                       & 14                       & 0.81                                  \n\\end{tabular}\n    \\caption{Printer head algorithms results variance}\n    \\label{tab:printer-table}\n\\end{table}\n\n\\subsubsection{Question 1}\nIn \\autoref{fig:printer-results} and \\autoref{fig:q1-printer-results} we observe that by enabling the preserve option, all algorithms consistently outperform their default variants. Therefore, we conclude that the answer for \\textbf{Q1} is \\textbf{yes}. This result confirms our expectation that constraint preservation will result in at least a minor improvement in all use cases.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Preserve comparison of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:q1-printer-results}\n\\end{figure}\n\\subsubsection{Question 2}\nBy analysing \\autoref{fig:q2-printer-results} and \\autoref{fig:printer-results} we observe that the reset approaches perform slightly worse when they have to perform reset operations quite often (solving simple tasks repeatedly), but end up performing similarly to the state-of-the-art approach in the long run. Therefore, we determine that the answer for \\textbf{Q2} is \\textbf{no}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for printer data during the first 100 seconds (showing standard error)}\n    \\label{fig:q2-printer-results}\n\\end{figure}\n\\subsubsection{Question 3}\nWe do not observe any significant benefit of the priority approaches that can not be attributed to noise. Therefore, we conclude that the answer for \\textbf{Q3} is \\textbf{no}. It is worth mentioning that both priority algorithms compute the most difficult task faster on average than the other approaches. Their average times to learn 13 tasks are 661 and 671 seconds, whereas the state-of-the-art approach achieves a mean of 1951 seconds and the best reset approach 1096. Is it highly likely that these results are a consequence of variation in the initial ordering and do not reflect the actual performance of the search strategies.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-BFS PRESERVE,\n    PRIO-EX PRESERVE, PRIO-CONS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Zoom in of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:q3-printer-results}\n\\end{figure}\n\\subsection{String transformation}\n\\subsubsection{Overview}\nIn \\autoref{fig:string-results} we present an overview of the performance of each algorithm, expressed as the percentage of tasks solved after several minutes. We also outline the variance in \\autoref{tab:string-table}, by listing their worst and best performances, along with the standard deviation of the total number of tasks solved.\n\nWe observe the fact that no algorithm was able to solve all the tasks. We attribute this to the fact that we have used functional testing in order to validate if a generated program is correct, whereas the original experiment did not enable this option. Functional testing refers to evaluating predicates by disregarding their truth value and only considering their output \\cite{functional-definition}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of string transformation results (showing standard error)}\n    \\label{fig:string-results}\n\\end{figure}\n\n \\begin{table}[hbt!]\n \\centering\n\\begin{tabular}{c|c|c|c}\n        Algorithm & \\multicolumn{1}{l|}{Min} & \\multicolumn{1}{l|}{Max} & \\multicolumn{1}{l}{Standard deviation} \\\\ \\hline\nPRIO-CONS          & 13                       & 15                       & 0.54                                   \\\\\nPRIO-CONS PRESERVE & 14                       & 16                       & 0.83                                    \\\\\nID PRESERVE      & 12                       & 19                       & 2.5                                    \\\\\nNAIVE              & 0                        & 21                       & 6.28                                   \\\\\nID               & 12                       & 21                       & 2.45                                   \\\\\nRESET-BFS          & 11                       & 38                       & 11.41                                  \\\\\nRESET-ID           & 13                       & 39                       & 9.78                                   \\\\\nPRIO-EX            & 20                       & 39                       & 7.87                                   \\\\\nRESET-BFS PRESERVE & 15                       & 40                       & 10.39                                  \\\\\nPRIO-EX PRESERVE   & 35                       & 40                       & 1.5                                    \n\\end{tabular}\n    \\caption{String transformation algorithms results variance\n    }\n    \\label{tab:string-table}\n\\end{table}\n\\subsubsection{Question 1}\nFor the first question, we compare the results of the three main search strategies with the constraint preservation option both enabled and disabled. The result of the comparison is present in \\autoref{fig:q1-string-results}. We conclude that the answer to \\textbf{Q1} is \\textbf{yes}, as all strategies perform at least as well as their default version when constraint preservation is used.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph showing the effects of the preserve option for string data (showing standard error)}\n    \\label{fig:q1-string-results}\n\\end{figure}\n\\subsubsection{Question 2}\nIn \\autoref{fig:q2-string-results} we compare the results of the state-of-the-art strategy, versus the two reset strategies we suggested. Since both reset strategies perform better than the state-of-the-art approach we conclude that the answer for \\textbf{Q2} is \\textbf{yes}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for string data (showing standard error)}\n    \\label{fig:q2-string-results}\n\\end{figure}\n\\subsubsection{Question 3}\nIn \\autoref{fig:q3-string-results} we observe the effect of using a priority ordering for the breadth first search approach. We have also plotted the performance of the state-of-the-art system to reference as a benchmark. The PRIO-EX algorithm, which sorts by the precision of the best program found until now, outperforms the original reset approach. On the other hand, the PRIO-CONS algorithm, which sorts by the number of constraints, exhibits below par performance. We associate this with the fact that the PRIO-CONS algorithm uses a poor heuristic. Tasks for which we have accumulated more constraints have a smaller hypothesis space size, but it is not a good indicator of being closer to a solution. \nFurthermore, by analysing \\autoref{tab:string-table} we observe that priority ordering results in a lower variance in the performance of the system. We consider this a possible upside, as it improves the system's reliability. \nWe conclude that the answer for \\textbf{Q3} is \\textbf{yes}, but we outline the need for further research in this area, with the purpose of developing more accurate heuristics.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-BFS PRESERVE, PRIO-EX PRESERVE, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph showing the effects of priority ordering for string data (showing standard error)}\n    \\label{fig:q3-string-results}\n\\end{figure}\n\\subsection{Robot movement}\nThis data set contains single examples for each task because any positive examples generated would be part of the same equivalence class (e.g. moving two cells to the right is independent of starting position). Consequently, there would be no benefit in testing the PRIO-EX algorithm against this data set. Moreover, the previous two data sets have shown that the PRIO-CONS approach performs poorly in practice. Therefore, we decide to conclude our answer for \\textbf{Q3} based on the previous data and only test our claims for \\textbf{Q1} and \\textbf{Q2} against this data set.\n\\subsubsection{Overview}\nIn \\autoref{fig:robot-results} we present an overview of the performance of each algorithm, expressed as the percentage of tasks solved after several minutes.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 75,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma,y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma, y error = SOTA PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of robot results (showing standard error)}\n    \\label{fig:robot-results}\n\\end{figure}\n\\subsubsection{Question 1}\nBy reviewing \\autoref{fig:robot-results}, we observe that using constraint preservation has improved the performance of two out of three of the tested strategies. As expected, RESET-BFS benefits the most from constraint preservation. On the other hand, the RESET-ID approach performs slightly worse when constraint preservation is enabled. By manually analysing the results of the data, we associate this finding with noise generated by order randomisation. Therefore, the answer for \\textbf{Q1} is \\textbf{yes}.\n\\subsubsection{Question 2}\n\\autoref{fig:q2-robot-results} showcases a comparison between the state-of-the-art approach and the two reset strategies. The current state-of-the-art approach solves the first few tasks much quicker because it does not reset the hypothesis space size on each found solution. After the 20 minutes mark, we observe that the state-of-the-art approach stops making progress because it has reached a large enough search size to be able to find any more solutions in a reasonable time frame. Whereas both reset approaches show that they are making constant progress, with their learning rate remaining relatively constant\\footnote{As mentioned in the introduction, we expect to observe a slight decay in learning rate as more tasks are added to the BK.}. Although the learning time of the first tasks is lower for the reset strategies, we evaluate that the ability to learn a higher percentage of tasks is more important when evaluating the performance of a system. Therefore, we conclude that the answer for \\textbf{Q2} is \\textbf{yes}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 75,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma, y error = SOTA PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for robot data (showing standard error)}\n    \\label{fig:q2-robot-results}\n\\end{figure}\n\\chapter{Conclusions}\n\\section{Summary}\nAlthough an area that has shown high potential, there has been little development in terms of multi-task learning approaches in ILP. We have experimented with refining the current state-of-the-art approach with the expectation of obtaining substantially improved performance. We presented our assumptions' theoretical basis and then empirically confirmed them against three data sets. We have implemented both our new approach and the state-of-the-art technique based on the Popper ILP system. Our experimental results show that (1) constraint preservation improves learning speed in multi-task settings (\\textbf{yes} to \\textbf{Q1}), (2) the benefits of search size reset overcome it's disadvantages (\\textbf{yes} to \\textbf{Q2}), and (3) heuristic ordering of programs can improve learning rate (\\textbf{yes} to \\textbf{Q3}). \n\nIn the following sections, we present the limitations of our work and suggest future work that could improve the performance of Popper in multi-task environments. \n\\section{Limitations}\nWe have identified multiple factors that restricted the scope of our work.\n\\paragraph{Limited data.} The use of multi-task learning is relatively new to the field of ILP. Therefore, we highlight the lack of publicly available data sets used to benchmark such systems. We believe that our findings would have been better supported by testing in a wider variety of data, which have been previously found to uncover more use cases we might not have considered in our project.\n\\paragraph{Implementation improvement.} Our implementation of multi-task Popper has one specific drawback we have encountered during testing. On each reset operation, the underlying ILP environment is re-initialised, which often results in long reset times. There exists a simple solution to this issue, which would enforce the system always to preserve constraints between runs or to iterate through the constraint list and ``forget\" each of them one by one. Such an approach would have introduced bias in our experimental results by favouring constraint preservation. We argue that a more specialised implementation of Popper would circumvent this problem and further improve learning time (especially for the reset approach, where this part of the code is called more frequently).\n\\paragraph{Bias settings.} One significant challenge in ILP is selecting bias parameters \\cite{ade1995declarative}. Examples of bias parameters are the maximum number of variables allowed in a program or the maximum number of clauses allowed in a program. Appropriate selection of bias parameters has a significant impact on the learning rate by modifying the size of the possible hypothesis space. We argue that in the case of multi-task learning, the initial bias, which was set by the user with the initial BK in mind, may no longer be appropriate. For example, some tasks may have three clause solutions when solved by themselves, but only two clause solutions in the multi-task learning environment. Therefore, if our system could automatically compute bias parameters, it would improve the learning rate by constraining the hypothesis spaces size when the BK is augmented.\n\\section{Future work}\n\\paragraph{Explainable results.} Without multi-task learning, each program learnt is built from primitives, which are given in the BK and known to a human operator. When multi-task learning is introduced, program definitions may also contain other programs. This affects the ability of a human to understand the programs in two ways. Firstly, we note that, in a multi-task learning setting, knowledge of all the new predicates that appear in a solution is required before reading it. Secondly, we outline the fact that it is a common occurrence for more complicated tasks to be selected in favour of simpler tasks when they cannot be differentiated. For example see \\autoref{fig:explainable-anc\/Results-example}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{minted}{prolog}\n    f0(A,B) :- draw1(A,B)\n    f0(A,B) :- move_right(A,B),f0(A,B)\n   \n   \n    f1(A,B) :- f0(A,B) \n   \n    f1(A,B) :- draw1(A,B)\n\\end{minted}\n\\caption{Example of two different ways in which a program can be learnt}\n\\label{fig:explainable-anc\/Results-example}\n\\end{figure}\n\nIn the above case, both \\texttt{draw1} and \\texttt{f0} appear to behave identically for the system, in the absence of negative examples. In consequence, we propose the development of a predicate translator. A predicate translator would take the output of a multi-task ILP system and transform it such that all predicates will only make use of definitions from the BK, and any clause that does not affect the correctness of the task will be pruned out. \n\\paragraph{Task ordering.} In this project, we have shown the potential of task ordering in improving the learning capacity of multi-task ILP systems. We suggest further research in developing more accurate heuristics to predict which tasks are closer to being solved. One of the solutions we propose is the use of other Machine Learning techniques in order to identify more manageable tasks and give them priority. We also concede the fact that there are certain limitations to the applicability of such an approach, especially when there are very few examples given for each task. \n\\paragraph{Parallel computation.} When searching for solutions of a specific size, one could modify the current approach that attempts to solve tasks sequentially to try and solve multiple tasks concurrently, in a bag of tasks manner. This method would allow one to vastly increase the learning rate of multi-task Popper on multi-core systems. Moreover, a significant bottleneck observed during all experiments is that the system got stuck searching for solutions to unsolvable tasks. This issue will be mitigated, as data sets would require as many high difficulty tasks as threads to congest the system. \n\\newpage\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}