| { | |
| "paper_id": "W89-0205", | |
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| "date_generated": "2023-01-19T03:44:53.903587Z" | |
| }, | |
| "title": "Head-Driven Bidirectional Parsing: A Tabular Method Giorgio Satta (0)(\u00b0\u00b0), Oliviero Stock (\u00b0\u00b0)", | |
| "authors": [], | |
| "year": "", | |
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| "abstract": "", | |
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| "abstract": [], | |
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| "text": "Tabular methods for context-free language analysis [Graham and Harrison, 1976, Graham et al., 1980] , and in particular Earley's Algorithm [Earley, 1970] , can be considered a major reference for natural language parsing. Even if independently con ceived , Earley's Algorithm constitutes the basis for Chart parsing [Kay, 1980 , Kaplan, 1973 . Som e recent works in the framework o f Chart parsing [Steel and D e R oeck, 1987, Stock et al., 1989] have pointed out the importance of bidirectionality for natural language analysis. Another work that deals with some form o f bidirectionality [B ossi et al., 1983] can be found in the form al language literature, though the analysis given there presupposes Chomsky normal form grammars.", | |
| "cite_spans": [ | |
| { | |
| "start": 51, | |
| "end": 62, | |
| "text": "[Graham and", | |
| "ref_id": "BIBREF4" | |
| }, | |
| { | |
| "start": 63, | |
| "end": 99, | |
| "text": "Harrison, 1976, Graham et al., 1980]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 139, | |
| "end": 153, | |
| "text": "[Earley, 1970]", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 316, | |
| "end": 326, | |
| "text": "[Kay, 1980", | |
| "ref_id": "BIBREF6" | |
| }, | |
| { | |
| "start": 327, | |
| "end": 341, | |
| "text": ", Kaplan, 1973", | |
| "ref_id": "BIBREF5" | |
| }, | |
| { | |
| "start": 398, | |
| "end": 408, | |
| "text": "[Steel and", | |
| "ref_id": "BIBREF10" | |
| }, | |
| { | |
| "start": 409, | |
| "end": 446, | |
| "text": "D e R oeck, 1987, Stock et al., 1989]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 590, | |
| "end": 611, | |
| "text": "[B ossi et al., 1983]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "In this paper w e shall introduce a tabular method coinceived for bidirectional contextfree parsing, discuss som e o f its relevant properties and through an exam ple give an idea o f how the algorithm works.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "Introduction", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "Assum e -(s, i,j), with j=[p, 0, K(p N = (S , A, B }, L={a, b, c, d, e) , and P is the production set given as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 6, | |
| "end": 36, | |
| "text": "e -(s, i,j), with j=[p, 0, K(p", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 37, | |
| "end": 71, | |
| "text": "N = (S , A, B }, L={a, b, c, d, e)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Definitions", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "1: ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definitions", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "S -\u00bb A", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definitions", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "Intemational Parsing Workshop '89", | |
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| "back_matter": [], | |
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| "FIGREF0": { | |
| "text": "One basic aspect o f known tabular m ethods, i.e. that the analysis proceedes m onodirectionally, is a relevant lim itation, that, although reasonable for artificial languages, seem s reductive for natural language. A strong reason for a bidirectional approach within natural language analysis is that modem theories o f grammar em phasize the role o f a particular elem ent inside each constituent (phrase), called the head; this element carries categorial as w ell as thematic information about other elements within the constituent. It turns out that the acceptability and the general skeleton o f each constituent, crucially depend on such information. More concretely, a number o f possible partial interpretations would be pruned out earlier, on the basis o f functional information attached to the head, resulting in greater efficiency.", | |
| "type_str": "figure", | |
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| "text": "e a context-free grammar G =(N , E, P, S), where N is the finite set o f all non terminal sym bols, Z is the set o f terminal sym bols, P is a finite set o f productions, and S e N is the start sym bol. L(G) represents the language generated by the grammar G. The productions in P are numbered from 1 to IPI1, and are all o f form D p-\u00bb C p j ... Cp ^) , where k is a function defined over the set {1 ... IP!} and that takes values in the set Z* (the set o f positive integers). In the following, the natural number p often will be used instead o f the production associated with it. Without loss of generality, here it is assumed that the grammar G is in e-free form (see [A ho and U llm an, 1972:147]); a more general formulation o f the algorithm does not lead to the loss o f the properties shown here. A function x is defined over the set { 1...IPI} and it takes values in Z+. This function indicates, for every production p in P, a position in the right-hand side o f the production, occupied by a sym bol in N u l . This position is called the head position, and the corresponding sym bol is said to be in the head position for production p. Every time, during the analysis, a symbol is recognized that is in head position for some production p , the presence o f the symbol D p relative to production p is then locally hypothesized DEFINITION 2.1 A sta te is d efin ed to be any quadruple [p, I d o t, rd o t, m ], with l< p^lP I , 0<ldot<rdot<K(p), me { Im, rm). The component p indicates the corresponding production in P; the components Idot and rdot represent two distinct positions, one after the other, in the right-hand side o f production p. The com ponent m is a simple indicator m -lm indicates that the value o f Idot cannot be further diminished, even if greater than zero, while m -rm indicates that the value o f rdot cannot be increased further, even if it is less than 7z(p). N ote that, by definition, one limitation excludes the other. The value is used for the indicator m in the absence o f both the limitations just described. The use o f the index m, as it w ill be shown, prevents the duplication o f \"partial analyses\" for substrings o f w. Every state j= [p , Idot, rdot, m] may be understood to be a partial analysis relative to production p, for w hich the constituents Cpjdot+1 ... Cp/dot* belonging to the right-hand side, have been recognized. In the follow ing, for convenience, the states w ill often be referred to in these terms. The sym bol Is denotes the set o f all states. contains all the states indicating partial analyses o f productions in which the sym bol X occupies the head position. DEFINITION 2.3 An equivalence relation Q,in Isx ls is defined so that for two generic states s=[p, Idot, r d o t, m] and j ' = [ p \\ Idot', rd o t', m rj, sQ \u00a7' holds if and only if p -p \\ I d o t-ld o t' the set Is), and is initialized with as empty set. The presentation o f the recognition algorithm is preceded by a schematic illustration o f the computation involved. The algorithm inserts into the recognition matrix T each state s that indicates a partial analysis previously obtained for the generic substring jWj. There is a one to one correspondence between the indicies o f the analyzed substring jWj and the indices o f the com ponent ry, in which state s has been inserted. The algorithm then processes each state, com bining it with nearby states in an effort to extend the portions o f the string dominated by these states. When the analysis relative to a particular state is com pleted (for both the right and left sides), if the constituent obtained is in a head position for som e production p in P, a new partial analysis for the production p itself is inserted into matrix T. Note that the algorithm straightforwardly separates the problem o f the combination o f different states from the problem o f control. The algorithm in fact does not specify the order in which the different states must be considered, nor in which order every single state must be expanded in the two opposing sides. To that end, the algorithm uses a variable A which takes values in the set ^(IsxN xN ). ALGORITHM 3.1 G iven a context-free grammar G =(N , E, P, S) in e-free form, let w=a\\ ... an, n>0 , be an input string. D evelop a recognition matrix T, o f size (n + l)x (rt+ l), w hose components fy are coindexed from 0 to n for both sides. triple e=(s, M , /) to set A only if sQsq does not hold for any triple e q = (sq, i -l , i) 4. while A not em pty do 2 A derivation tree D associated with a string L(G), is a labeled tree formed by all the productions used in the derivation ofw, representing the correct hierarchic order. The procedure is applied only when e -{s , iyJ) with s=[p, Idot, rdot, rdot, -] is created and the triple e'=(s\", i \\ j ) is insened in set A, only if j\"Q^q does not hold for any state in or for any triple \u00a3q=(5q, i \\ j ) in A. If at least one state s ' is found with the above properties, set m -rm in s. Case 2: C p^e l . If Cp ^cjot=ai, the state s'=[p, ld o t-\\y rdot, -] is created and the triple e'=(s\\ i-1, j) is insened into set A, only if j'C&q does not hold for any state s in t{ j or for any triple ^q=(5q, i-l,j) in A. If C^^Q=a{, set m=rm in 5. This procedure is applied only if state s can be extended leftward (ldot>0) and only if it has not already been extended rightward that is, if it is not subsumed to the right by a more updated state. There are two cases, depending upon whether the left-hand expansion sym bol is a terminal sym bol or not. If Cp ldol is a non-terminal sym bol, the search proceeds to the left o f state j, to any state s' (adjacent), that corresponds to a com pleted analysis rdot'=n(pr)) o f a constituent usable by state s (Dp.=Cpjdot)* If successful, the analysis is extended in correspondence with state s, including the constituent found nearby; state s then is marked with m -r m , since this has been subsumed on the left by a more updated state. If Cp Jdol is, instead, a terminal sym bol, and if C an extension o f the analyses corresponding to state s is made, including the terminal sym bol a^ Still, state s is marked with m -rm for the same reasons as if j\"C\u00a3q does not hold for any state sq in t^, or for any triple eq=Csq, i j l in A. If at least one state s' has been found with the properties described above, set m-lm in s. Case 2: Cp rdot+1e I . If Cp rdot+1=aj+1, the state s'=[p, Idot, rdot+l, -] is created and the triple e'=(s\\ i j + l ) is inserted in A, only if .s'Q?q does not hold for any state s q in riJ+ r 15 c P,idot= aj + r set m=lm in s -This procedure is symmetric to the left-expander procedure, so the explanation is omitted. PROCEDURE 3.3 Left-completer Precondition The procedure is applied only when e-{s, /,/) , with J=[p, 0 , 7i(p), o t'+ 1, -] is created and the triple e'-{s\", i \\ j ) is inserted in set A only if s\"Q$q does not hold for any state in r-,j or for any triple e = (jq, i \\ j ) in A. Furthermore, set m'-lm for every s' found. This procedure is applied w henever the analysis o f a constituent D p has been com pleted through a state s=[p, 0, Kip), m]. It proceeds by searching leftward o f state j for any adjacent state s' that has not yet been subsumed to the left (m'*rm) and is able to \"exp an d \" state 5 CDp= C p. rdot.+1). If su ccessfu l, an exten sion o f the analysis corresponding to s' is carried out, including the constituent D p. State s' is then marked with m-lm, since it has now been subsumed on the right by a more updated state. set A only if .y\"C&q does not hold true for any state sq in or for any triple eq=(>yq, y\") in A. Furthermore, set m -r m for every s' found.This procedure is symmetric to the left-completer procedure, so the explanation", | |
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| "FIGREF2": { | |
| "text": "), m\\. Description For every s e F(D p), insert the triple e=(s, i, j) in set A only if sQ?q does not hold for any state sq in t[j or for any triple eq=(sq, i, j ) in A.The procedure is applied whenever the analysis of a particular constituent has been com pleted and this constituent occupies the head position in some production p. In this case a new state corresponding to a partial analysis for production p is created, including the head. Once again, note that the procedure never duplicates triples in A, nor states belonging to the same component of the recognition matrix T.4. Some Formal Properties of the AlgorithmIn this section the most interesting properties o f Algorithm 3.1 are stated. For a formal proof o f what follow s refer to [Satta and Stock, 1989b]. Four major properties have been grouped under Invariant 4.1 below. Note that soundness and com pleteness for Algorithm 3.1 follow straightforwardly from statements (i) and (ii) in Invariant 4.1.", | |
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| "FIGREF3": { | |
| "text": "+1 <x(p)<rdot only if a quadruple h=[h\\, hi, /13, h4\\, hq>0, 1^7<4 exists such that s=[p, ldot-h\\, rdot-^h^, m]e ti_h3j+h( s the extension o f a state to both the left and right sides. This possibility, if not carefully controlled, can lead to the duplication o f an analysis, in the follow in g w ay. If a state s , relative to a partial analysis for a constituent Cs, is independently extended to both sides, it would lead to the introduction o f tw o partially overlapping states, s' and s'\\ for the same analysis. The com pletion o f s' and 5\" then w ould lead to the duplication o f constituent Cs-The algorithm presented here uses the index m, associated with each state, so as to avoid partial overlapping for two (states s=[p, Idot, rdot, m ]e ry and 5 = [p , Idot', r d o t\\ t[' j ' are pa rtially overlapped (s' Dsr) iff /< /'< /< /, ldot<ldot'<rdot<rdot\\ and, furthermore, s subsum es the same constituents Cpjdot'+l-Cpjciot subsumed in s '. N ote that for two states s=[p, Idot, rdot, m ] and J = [ p , Idot', rd o t\\ m *] such that s(Ds\\ it always holds that Idot'<z(p)<rdot. The follow ing theorem can now be stated.", | |
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| "FIGREF5": { | |
| "text": "the string w=cceccb is sim plified by the follow ing steps (the order o f application for the five procedures at line 5 is chosen at random). inserted in f 1>4 and m is set to Im in state s4 , [5 , 1,2,-] is inserted in r1>4 by the trigger procedure; 6) j 7=[5, 1,3,-] is inserted in and m is set to Im in state by Case 2 o f the right-expander procedure; 1 , 2 ,-] is inserted in r0,5 by the trigger procedure; setting o f the m components in states 53 and 55 prevents the expansion of partial analysis at both sides. Though not shown here, in more com plicated cases the setting o f the m components permits the left-completer procedure to combine a state s with the \"leftward largest\" partial analyses that are adjacent to the left o f s, preventing once more partial analysis duplication (vice versa for the right-completer procedure). a parsing algorithm that extends bidirectionally the classic tabular methods for context-free language analysis. The algorithm is given for e-free form context-free grammars, but it is not difficult to extend it to the general case, for exam ple by em ploying the same technique used in [Graham et al. 1980] in the treatment o f empty categories. With respect to natural language parsing, the presented tabular method is compatible with the well known \"Active Chart Parsing\" technique, as pointed out in [Satta and Stock 1989a]. Finally, the extension to Earley's Algorithm proposed in [Shieber 1985] for parsing com plex-feature-based formalisms, could be equally applicable to the presented approach.", | |
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