| { | |
| "paper_id": "1993", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T07:36:44.286340Z" | |
| }, | |
| "title": "Stochastic Lexicalized Context-Free Grammar", | |
| "authors": [ | |
| { | |
| "first": "Yves", | |
| "middle": [], | |
| "last": "Schabes", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "Mitsubishi Electric Research Laboratories", | |
| "institution": "", | |
| "location": { | |
| "addrLine": "Cambrid \ufffd e", | |
| "postCode": "201, 02139", | |
| "settlement": "Broadway", | |
| "region": "MA" | |
| } | |
| }, | |
| "email": "" | |
| }, | |
| { | |
| "first": "Richard", | |
| "middle": [ | |
| "C" | |
| ], | |
| "last": "Waters", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "Mitsubishi Electric Research Laboratories", | |
| "institution": "", | |
| "location": { | |
| "addrLine": "Cambrid \ufffd e", | |
| "postCode": "201, 02139", | |
| "settlement": "Broadway", | |
| "region": "MA" | |
| } | |
| }, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "Stochastic lexicalized context-free grammar (SLCFG) is an attractive compromise between the parsing efficiency of stochastic context-free grammar (SCFG) and the lexical sensitivity of stochas tic lexicalized tree-adjoining grammar (SLTAG). SLCFG is a restricted form of SLTAG that can only generate context-free languages and can be parsed in cubic time. However, SLCFG retains the lexical sensitivity of SLTAG and is therefore a much better basis for capturing distributional information about words than SCFG.", | |
| "pdf_parse": { | |
| "paper_id": "1993", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "Stochastic lexicalized context-free grammar (SLCFG) is an attractive compromise between the parsing efficiency of stochastic context-free grammar (SCFG) and the lexical sensitivity of stochas tic lexicalized tree-adjoining grammar (SLTAG). SLCFG is a restricted form of SLTAG that can only generate context-free languages and can be parsed in cubic time. However, SLCFG retains the lexical sensitivity of SLTAG and is therefore a much better basis for capturing distributional information about words than SCFG.", | |
| "cite_spans": [], | |
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| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "Recently, it has been suggested that stochastic lexicalized tree-adjoining grammar (SLTAG) [8, 9] ", | |
| "cite_spans": [ | |
| { | |
| "start": 91, | |
| "end": 94, | |
| "text": "[8,", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 95, | |
| "end": 97, | |
| "text": "9]", | |
| "ref_id": "BIBREF8" | |
| } | |
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| "section": "The application of stochastic techniques to syntax modeling has recently regained popularity. Most of the work in this area has tended to empha size one or the other of the following two goals. The first goal is to capture as much distributional information about words as possible. The second goal is to capture as many of the hierarchical con straints inherent in natural languages as possible. Unfortunately, these two goals have been more or less incompatible to date. Early stochastic proposals such as Markov Models, N-gram models [2, 14] and Hidden Markov Models [7] are very effective at captur ing simple distributional information about adja cent words. However, they cannot capture long range distributional information nor the hierar chical constraints inherent in natural languages. Stochastic context-free grammar (SCFG) [1, 3, 5] extends context-free grammar (CFG) by as sociating each rule with a probability that con trols its use. Each rule is associated with a single probability that is the same for all the sites where the rule can be applied. SCFG captures hierarchical information just as well as CFG; however , it does not do a good job of capturing distributional information about words. There are at least two reasons for this. First, many rules do not contain any words and therefore the associated probabilities do not have any direct link to words. Second, distributional phenomena that involve the application of two or more rules do not have a direct link to any of the stochastic parameters of SCFG, because the probabilities apply only to single rules. It has been observed in practice that SCFG performs worse than non-hierarchical approaches. This has lead many researchers to believe that simple distributional information about adjacent words is the most important single source of in formation. In the absence of a formalism that adequately combines this information with other kinds of information, the emphasis in research has been on simple non-hierarchical statistical models of words, such as word N-gram models.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Unfortunately, the statistical algorithms for SLTAG [9] require much more computational re sources than the ones for SCFG. For instance, the algorithms for estimating the stochastic parame ters and determining the probability of a string require in the worst case O(n 6 )-time for SLTAG [9] but only O(n 3 )-time for SCFG [3] . increases the trees that can be generated without increasing the ambiguity of derivations.)", | |
| "cite_spans": [ | |
| { | |
| "start": 52, | |
| "end": 55, | |
| "text": "[9]", | |
| "ref_id": "BIBREF8" | |
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| "start": 287, | |
| "end": 290, | |
| "text": "[9]", | |
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| "start": 322, | |
| "end": 325, | |
| "text": "[3]", | |
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| "section": "may be able to capture both distributional and hierarchical information. An SLTAG grammar consists of a set of trees each of which contains one or more lexical items. These elementary trees can be viewed as the elementary clauses (including their transformational variants) in which the lex ical items participate. The elementary trees are combined by substitution and adjunction. Each possible way of combining two trees is associated with a probability. Since it is based on tree-adjoining grammar (TAG), SLTAG can capture some kinds of hier-archical information that cannot be captured by SCFG. However, the key point of comparison be tween SLTAG and SCFG is that since SLTAG is lexicalized and uses separate probabilities gov erning each possible combination of trees, each probability is directly linked to a pair of words. This makes it possible to represent a great deal of distributional information about words.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The only important difference between LCFG and LTAG is that LTAG allows both elementary and derived wrapping auxiliary trees. The im portance of this is that wrapping adjunction (see Figure 2d ) encodes string wrapping and is there fore context sensitive in nature. In contrast, left and right adjunction (see Figures 2b & 2c ) merely support string concatenation. As a result, while LTAG is context sensitive in nature, LCFG is lim ited to generating only context-free languages.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 183, | |
| "end": 192, | |
| "text": "Figure 2d", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 310, | |
| "end": 325, | |
| "text": "Figures 2b & 2c", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "To see that LCFG can only generate context free languages, consider that any LCFG G can be converted into a CFG generating the same strings in two steps as follows. First, G is converted into a tree substitution grammar (TSG) G' that gen erates the same strings. Then, this TSG is con verted into a CFG G\". A TSG is the same as an LCFG (or LTAG) exc\ufffdpt' that there cannot be any auxiliary trees. To create G' first make every initial tree of G be an initial tree of G' . Next, make every auxiliary tree T of G be an initial tree of G' . When doing this, relabel the foot of T with c ( turning T into an initial tree). In addition, let A be the label of the root of T. If T is a left auxiliary tree, rename the root to AL; otherwise rename it to AR,", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "To complete the creation of G' alter every node TJ in every initial tree in G' as follows: Let A be the label of T/\u2022 If left adjunction is possible at T/, add a new first child of T/ labeled AL , mark it for substitution, and add a tree corresponding to A L --+ c if one does not already exist. Right ad-junction is handled analogously by adding a new last child of T/ labeled A R and insuring the exis tance of a tree corresponding to A R --+ c.", | |
| "cite_spans": [], | |
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| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "The TSG G' generates the same strings as G, because all cases of adjunction have been changed into equivalent substitutions. Note that the transformation would not work if LCFG al lowed wrapping auxiliary trees. The TSG G' can be converted into a CFG G\" by flattening each tree in G' into a context-free rule that expands the root of the tree into the frontier in one step.", | |
| "cite_spans": [], | |
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| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Although the string sets generated by LCFG are the same as those generated by CFG, LCFG is capable of generating more complex sets of trees than CFG. In particular, it is interesting to look at the path sets of the trees generated. (The path set of a grammar is the set of all paths from root to frontier in the trees generated by the grammar. The path set is a set of strings over \ufffd U NT U { c}.)", | |
| "cite_spans": [], | |
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| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "The path sets for CFG ( and TSG) are regu lar languages [15] . In contrast, just as for LTAG and TAG, the path sets for LCFG are context-free languages. To see this, consider that adjunction makes it possible to embed a sequence of nodes ( the spine of the auxiliary tree) in place of a node on a path. Therefore, from the perspective of the path set, auxiliary trees are analogous to context free productions. Figure 3 summarizes the relationship be tween LCFG and several other grammar for malisms. The horizontal axis shows the com plexity of strings that can be generated by the formalisms, i.e., regular languages (RL), context free languages (CFL), and tree adjoining lan guages (TAL). The vertical axis shows the com plexity of the path sets that can be generated. ", | |
| "cite_spans": [ | |
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| "start": 56, | |
| "end": 60, | |
| "text": "[15]", | |
| "ref_id": "BIBREF15" | |
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| "ref_spans": [ | |
| { | |
| "start": 411, | |
| "end": 419, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Comparisons", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "As shown in [12, 13] LCFG lexicalizes CFG with out changing the trees derived. Further, a con structive procedure exists for converting any CFG G into an equivalent LCFG G'.", | |
| "cite_spans": [ | |
| { | |
| "start": 12, | |
| "end": 16, | |
| "text": "[12,", | |
| "ref_id": "BIBREF11" | |
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| "start": 17, | |
| "end": 20, | |
| "text": "13]", | |
| "ref_id": "BIBREF13" | |
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| "section": "LCFG lexicalizes CFG", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "The fact that LCFG lexicalizes CFG is signifi cant, because every other method for lexicalizing CFGs without changing the trees derived requires context-sensitive operations [4] and therefore dra matically increases worst case processing time.", | |
| "cite_spans": [ | |
| { | |
| "start": 174, | |
| "end": 177, | |
| "text": "[4]", | |
| "ref_id": "BIBREF3" | |
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| "section": "LCFG lexicalizes CFG", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "As shown in [12, 13] (and in Section 4) LCFG can be parsed in the worst case just as quickly as CFG. Since LCFG is lexicalized, it is expected that it can be parsed much faster than CFG in the typical case.", | |
| "cite_spans": [ | |
| { | |
| "start": 12, | |
| "end": 16, | |
| "text": "[12,", | |
| "ref_id": "BIBREF11" | |
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| "end": 20, | |
| "text": "13]", | |
| "ref_id": "BIBREF13" | |
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| "section": "LCFG lexicalizes CFG", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "The definition of stochastic lexicalized context free grammar (SLCFG) is the same as the defini tion of LCFG except that probabilities are added that control the combination of trees by adj unc tion and substitution.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Stochastic LCFG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "For every root p of an initial tree, P 1 (p) is the probability that a derivation starts with the tree rooted at p. It is required that:", | |
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| "section": "Definition 2 An SL CFG is an 11-tuple (E,NT, I,A,S,P1,Ps ,PL,PN L,Pn ,PNR), where (E,NT,I, A,S) is an LCFG and Pi , Ps , PL , PN L, PR , and PN R are statis_tical parameters as defined below.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ") = 1 p", | |
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| "ref_spans": [], | |
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| "section": "Note that P1 (p) # 0 if and only if p is labeled S. For every root p of an initial tree and every node 'TJ that is marked for substitution, Ps (p, 'TJ) is the probability of substituting the tree rooted at p for 'T/ \u2022 For each 'T/ it is required that: For every node TJ in every elementary tree, PN L ( 'TJ) is the probability that left adjunction will not occur on 'T/ \u2022 For every root p of a left auxiliary tree, PL (p, 'TJ) is the probability of adjoining the tree rooted at p on 'T/ \u2022 For each 'T/ it is required that: PN L('TJ) + L PL (P,'TJ", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The probability of a string is the sum of the probabilities of all the different ways of deriving it. A most likely derivation of a string is a deriva tion that has as large a probability as any other derivation for the string. The probability of a tree generated by an SLCFG for a string is the sum of the probabilities of every way of deriving the tree. (Unlike in SCFG, in SLCFG there .ean be more than one way to derive a given tree.) A mo_ st likely tr.ee generated for a string is a tree whose probability is as large as any other tree generated for the string. (Note that a most likely derivation need not generate a most likely tree.)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "PN L ( 'TJ) = 0 if and only if left adjunction on 'T/ is obligatory. The parameters PN R('TJ) and PR (P, 1J) control right adjunction in an exactly analogous way. An SLCFG derivation is described by the ini tial tree it starts with, together with the sequence of substitution and adjunction operations that take place. The probability of a derivation is defined as the product of: the probability P 1 of starting with the given tree, the probabilities Ps , PL , and . PR of the operations that occurred, and the probabilities PN L and PN R of adjunction not occurring at the places where it did not occur.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Since SLCFG is a restricted case of SLTAG the ( 6 ) . ' 0 n -time SLTAG parser [9] can be used for parsing SLCFG. Further, it can be straightfor wardly modified to require at most O(n 4 )-time when applied to SLCFG. However, this does not take full advantage of the context-freeness of SLCFG.", | |
| "cite_spans": [ | |
| { | |
| "start": 46, | |
| "end": 51, | |
| "text": "( 6 )", | |
| "ref_id": "BIBREF5" | |
| }, | |
| { | |
| "start": 79, | |
| "end": 82, | |
| "text": "[9]", | |
| "ref_id": "BIBREF8" | |
| } | |
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| "section": "Parsing SLCFG", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "This section demonstrates that SLCFG can be parsed in O(n 3 )-time by exhibiting a CKY-style ley style LCFG recognizer presented in [12] .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 93, | |
| "end": 136, | |
| "text": "ley style LCFG recognizer presented in [12]", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Parsing SLCFG", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Ps = PL = PR = 0, PN L = Piv R = 1, and cru .:. dally P1 = 1. P1 = 0 for the other initial trees. G if and only if the proba bility of a1 \u2022 \u2022 \u2022 a n in G 77 is p without considering derivations where left (right) adjunction occurs on the original root of T'. ( Note that if 1J is a foot node, T' is an empty tree. The only string covered by 1J is the empty string; however, the empty string is covered with probability 1, because the empty string is the only string derived by G w ) 4", | |
| "cite_spans": [ | |
| { | |
| "start": 256, | |
| "end": 259, | |
| "text": "T'.", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 98, | |
| "end": 213, | |
| "text": "G if and only if the proba bility of a1 \u2022 \u2022 \u2022 a n in G 77 is p without considering derivations where left (right)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 262, | |
| "end": 487, | |
| "text": "Note that if 1J is a foot node, T' is an empty tree. The only string covered by 1J is the empty string; however, the empty string is covered with probability 1, because the empty string is the only string derived by G w )", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Suppose that G is an SLCFG and that a1 \u2022 \u2022 \u2022 a n is an input string. Let 1J be a node in an elementary tree (identified by the name of the tree and the position of the node in the tree). Label( 1J) E EU NT Uc is the label of the node. The predicate Is l nitialRoot( 1J) is true if and only if 1J is the root of an initial tree. Parent( 1J) is the node that is the parent of 1J or ..L if 1/ has no par ent. FirstChild(rJ) is the node that is the leftmost child of 1J or ..L if 1J has no children. Sibling(rJ) is the node that is the next child of the parent of 1J (in left to right order) or ..L if there is no such node. The predicate Substitutable(\u00b5, rJ) is true if and only if 1J is marked for substitution and p is the root of an initial tree that can be substituted for 1/\u2022 The predicate Radjoinable( p , rJ) is true if and only if p is the root of an elementary right aux iliary tree that can adjoin on 1/ \u2022 The predicate Ladjoinable( p , 1J) is true if and only if p is the root of an elementary left auxiliary tree that can adjoin on 1/ \u2022 The concept of covering is critical to the bottom-up algorithm shown below. Informally speaking, a node 7J covers a string if and only if the string can be derived starting from 1/ \u2022 More precisely, for every node 1J in every ele mentary tree in G, let T' be a copy of the subtree of T that is rooted at 1/ \u2022 Extend T' by adding a new root whose only child is the original root of T'. Label the new root of T' with a unique new symbol S'. If there is a node on the frontier of T' that is marked as the foot, relabel this node with c. This converts T' into an initial tree. Let a,, be an SLCFG that is identical to G except that .. T'. is\ufffdintroduced as an additional initial tree and the start symbol of G 77 is S' . The probabil ities associated with the the interior nodes of T' are identical to those for the corresponding nodes in T. The probabilities for the root of T' are", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "1J is labeled ai + l with probability one. The al gorithm then considers all possible ways of com bining matched substrings into longer matched substrings-it fills the upper diagonal portion of the array C[i, k] (0 \ufffd i \ufffd k \ufffd n) for increasing values of k -i.", | |
| "cite_spans": [], | |
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| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Two observations are central to the efficiency of this process. Since every auxiliary tree ( ele mentary and derived) in SLCFG is either a left or right auxiliary tree, the substring matched by a tree is always a contiguous string. Further, when matched substrings are combined, the al gorithm only has to consider adjacent substrings. (In SLTAG, a tree with a foot can match a pair of strings that are not contiguous-one left of the foot and one right of the foot.)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "There are three situations where combination of matched substrings is possible: sibling concate nation, left concatenation, and right concatena tion.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "As illustrated in Figure 4 , sibling concate nation combines the substrings matched by two sibling nodes into a substring matched by their parent. In particular, suppose that there is a node 1/o (labeled B in Figure 4 ) with two chil dren 1/i (labeled A) and TJ 2 (labeled A'). If Left concatenation (see Figure 5a ) combines the substring matched by a left auxiliary tree with the substring matched by a node the auxiliary tree can adjoin on. Right concatenation (see Fig ure 5b ) is analogous.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 18, | |
| "end": 26, | |
| "text": "Figure 4", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 209, | |
| "end": 217, | |
| "text": "Figure 4", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 305, | |
| "end": 314, | |
| "text": "Figure 5a", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 469, | |
| "end": 480, | |
| "text": "Fig ure 5b", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The algorithm (see Figure 6 ) is written in two parts: a main procedure Probability(a1 The main procedure repeatedly scans the ar ray C, building up longer and longer matched substrings until it determines all the S-rooted de rived trees that match the input. The purpose of the codes ( { L, R} etc.) is to insure that left and right adjunction can each be applied at most once on a node. The procedure could easily be modified to account for other constraints on the way derivation should proceed, such as those sug gested for LTAGs [11] . C[i, k] .", | |
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| "text": "[11]", | |
| "ref_id": "BIBREF10" | |
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| "end": 27, | |
| "text": "Figure 6", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 541, | |
| "end": 548, | |
| "text": "C[i, k]", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The procedure Add also propagates informa tion from one triple to another in situations where the length of the matched string is not increased-Le., when a node is the only child of its parent, when substitution occurs, and when adjunction is not performed.", | |
| "cite_spans": [], | |
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| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The O(n 3 ) complexity of the algorithm fol lows from the three nested induction loops on d, i and j. (Although the procedure Add is defined recursively, the number of pairs added to C is bounded by a constant that is independent of sen tence length.)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The algorithm does not depend on the fact that SLCFG is lexicalized-it would work equally well if were not lexicalized. If the sum p' + p on the third line of the Add procedure is changed to max(p' ,p) the algorithm computes the probabil ity of a most probable derivation. By keeping a record of every attempt to enter a triple into a cell of the array C, one can extend the algorithm so that derivations and therefore the trees they generate can be rapidly recovered.", | |
| "cite_spans": [], | |
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| "section": "The process starts by placing each foot node and each frontier node that is labeled with the empty string in every cell C[i, i] with probability one. This signifies that they each cover the empty string at all positions. The initialization also puts each terminal node 1/ in every cell C[ i, i + 1] where", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In the general case, the training algorithm for SCFG [5] requires O(n 3 )-time for each sentence of length n. A training algorithm for SLCFG can be constructed that achieves these same worst case bounds.", | |
| "cite_spans": [ | |
| { | |
| "start": 53, | |
| "end": 56, | |
| "text": "[5]", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
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| "section": "Training an SLCFG", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "To start with, since SLCFG is a restricted case of stochastic lexicalized tree-adjoining grammar (SLTAG), the O(n 6 )-time inside-outside reesti mation algorithm for SLTAG [9] can be used for estimating the parameters of an SLCFG given a training corpus. Straightforward modifications lead to an O(n 4 )-time algorithm for training an SLCFG. However, this alone does not achieve the full potential of SLCFG.", | |
| "cite_spans": [ | |
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| "start": 172, | |
| "end": 175, | |
| "text": "[9]", | |
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| "section": "Training an SLCFG", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "The same basic construction that underlies the algorithm in the last section can be used as the basis for an O(n 3 ) inside-outside training al gorithm for SLCFG. As in the last section, the key reason for this is that computations involving SLCFG only require the consideration of contigu ous strings.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Training an SLCFG", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "It should be noted that in the special case of a fully bracketed training corpus, the parameters of an SCFG can be estimated in linear time [6, 10] .", | |
| "cite_spans": [ | |
| { | |
| "start": 140, | |
| "end": 143, | |
| "text": "[6,", | |
| "ref_id": "BIBREF5" | |
| }, | |
| { | |
| "start": 144, | |
| "end": 147, | |
| "text": "10]", | |
| "ref_id": "BIBREF9" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Training an SLCFG", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "It is an open question whether this can be done for SLCFG. However, it should be straightforward to design an O(n 2 )-time training algorithm for SLCFG given a fully bracketed corpus.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Training an SLCFG", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "The preceding sections present stochastic lexi calized context-free grammar (SLCFG). SLCFG combines the processing speed of SCFG with the much greater ability of SLTAG to capture dis tributional information about words. As such, SLCFG has the potential of being a very useful tool for natural language processing tasks where statistical assessment/prediction is required.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusion", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "In (13) these three kinds of auxiliary trees are referred to differently a.s right recursive, left recursive, and centrally recursive, respectively.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": "for all frontier nodes rJ in AU J where Label(rJ) = c, Add('f/, 0, i, i, 1) for i = 0 to n -l for all frontier nodes rJ in AU J where Label(rJ) = ai + 1 , Add(TJ, 0, i, i + 1, 1) ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "annex", | |
| "sec_num": null | |
| } | |
| ], | |
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| "FIGREF0": { | |
| "uris": null, | |
| "text": "Example LCFG trees. Tree combination: (a) substitution, (b) left adjunction, (c) right adjunction, and (d) wrap ping adjunction, which is not allowed by SLCFG.", | |
| "type_str": "figure", | |
| "num": null | |
| }, | |
| "FIGREF1": { | |
| "uris": null, | |
| "text": "The tree and string complexity of LCFG and several other formalisms", | |
| "type_str": "figure", | |
| "num": null | |
| }, | |
| "FIGREF2": { | |
| "uris": null, | |
| "text": "1J1 ,{L, R},P1] E C[i,j] and [1J2, {L, R},P2] E C[j, k] then [rJo , 0,P1 x P2] E C[i, k].", | |
| "type_str": "figure", | |
| "num": null | |
| }, | |
| "FIGREF3": { | |
| "uris": null, | |
| "text": "\u2022\u2022\u2022an) and a subprocedure Add(rJ, code, i, k), which adds the triple [TJ, code,p] into C[i, k] .", | |
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| "num": null | |
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| "num": null, | |
| "content": "<table><tr><td colspan=\"2\">/\\ NP NPoJ. VP I\\ I\\ DJ. N V NP1J.</td><td>I\\ NPo VP I I Ei V</td><td>VP I\\ V VP\"'</td><td colspan=\"2\">N I\\ /\\ VP AN * VP\"' Adv</td><td>s /\\ NPoJ. VP I\\ V S 1 *NA</td></tr><tr><td>I boy</td><td>I saw</td><td>I left</td><td>I seems</td><td>I pretty</td><td>I smoothly</td><td>I think</td></tr></table>", | |
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| "text": ") create context-free lan guages, but the path sets they create are regular languages. LTAG and TAG generate tree adjoin ing languages and have path sets that are context free languages. LCFG is intermediate in nature. It can only generate context-free languages, but has path sets that are also context-free languages.", | |
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| "TABREF2": { | |
| "num": null, | |
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| "text": "The procedure Add enters a triple [TJ, al ready present in C[i, k], then the probability pto p' + p to reflect the fact that an ad ditional derivation of ai + l \u2022 \u2022 \u2022 ak has been found. Otherwise, a new triple [71, code, p] is added to", | |
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