{ "paper_id": "P99-1012", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:32:12.151948Z" }, "title": "Preserving Semantic Dependencies in Synchronous Tree Adjoining Grammar*", "authors": [ { "first": "William", "middle": [], "last": "Schuler", "suffix": "", "affiliation": { "laboratory": "", "institution": "University of Pennsylvania", "location": { "addrLine": "200 South 33rd Street Philadelphia", "postCode": "19104", "region": "PA", "country": "USA" } }, "email": "schuler@linc@cis.upenn" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "Rambow, Wier and Vijay-Shanker (Rainbow et al., 1995) point out the differences between TAG derivation structures and semantic or predicateargument dependencies, and Joshi and Vijay-Shanker (Joshi and Vijay-Shanker, 1999) describe a monotonic compositional semantics based on attachment order that represents the desired dependencies of a derivation without underspecifying predicate-argument relationships at any stage. In this paper, we apply the Joshi and Vijay-Shanker conception of compositional semantics to the problem of preserving semantic dependencies in Synchronous TAG translation (Shieber and Schabes, 1990; Abeill~ et al., 1990). In particular, we describe an algorithm to obtain the semantic dependencies on a TAG parse forest and construct a target derivation forest with isomorphic or locally non-isomorphic dependencies in O(n 7) time.", "pdf_parse": { "paper_id": "P99-1012", "_pdf_hash": "", "abstract": [ { "text": "Rambow, Wier and Vijay-Shanker (Rainbow et al., 1995) point out the differences between TAG derivation structures and semantic or predicateargument dependencies, and Joshi and Vijay-Shanker (Joshi and Vijay-Shanker, 1999) describe a monotonic compositional semantics based on attachment order that represents the desired dependencies of a derivation without underspecifying predicate-argument relationships at any stage. In this paper, we apply the Joshi and Vijay-Shanker conception of compositional semantics to the problem of preserving semantic dependencies in Synchronous TAG translation (Shieber and Schabes, 1990; Abeill~ et al., 1990). In particular, we describe an algorithm to obtain the semantic dependencies on a TAG parse forest and construct a target derivation forest with isomorphic or locally non-isomorphic dependencies in O(n 7) time.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "The primary goal of this paper is to solve the problem of preserving semantic dependencies in Isomorphic Synchronous Tree Adjoining Grammar (ISTAG) Shieber and Schabes, 1990 ), a variant of Tree Adjoining Grammar (Joshi, 1985) in which source and target elementary trees are assembled into isomorphic derivations. The problem, first described in Rambow, Wier and Vijay-Shanker (Rainbow et al., 1995) , stems from the fact that the TAG derivation structure -even using a flat adjunction of modifiers (Schabes and Shieber, 1994) -deviates from the appropriate dependency *The author would like to thank Karin Kipper, Aravind Joshi, Martha Palmer, Norm Badler, and the anonymous reviewers for their valuable comments. This work was partially supported by NSF Grant SBP~8920230 and ARO Grant DAAH0404-94-GE-0426. structure in certain cases. This can result in translation errors.", "cite_spans": [ { "start": 148, "end": 173, "text": "Shieber and Schabes, 1990", "ref_id": "BIBREF11" }, { "start": 213, "end": 226, "text": "(Joshi, 1985)", "ref_id": "BIBREF4" }, { "start": 346, "end": 399, "text": "Rambow, Wier and Vijay-Shanker (Rainbow et al., 1995)", "ref_id": null }, { "start": 499, "end": 526, "text": "(Schabes and Shieber, 1994)", "ref_id": "BIBREF9" }, { "start": 615, "end": 661, "text": "Aravind Joshi, Martha Palmer, Norm Badler, and", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "For example, if we parse sentence (1),", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "(1) X is supposed to be able to fly.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "using the trees in Figure 1 , we get the following derivation:l a:fly I 131 :be-able-to(VP) I j32:is-supposed-to (VP) with the auxiliary is-supposed-to adjoining at the VP to predicate over be-able-to and the auxiliary be-able-to adjoining at the VP to predicate over fly. If we then try to assemble an isomorphic tree in a language such as Portuguese (which makes less use of raising verbs) using the ISTAG transfer rules in Figure 2 , we will be forced into an ill-formed derivation:", "cite_spans": [ { "start": 113, "end": 117, "text": "(VP)", "ref_id": null } ], "ref_spans": [ { "start": 19, "end": 27, "text": "Figure 1", "ref_id": null }, { "start": 426, "end": 434, "text": "Figure 2", "ref_id": "FIGREF1" } ], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": ": voar", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": ";31 :~-capaz-de (VP) I /~2 :~-pressuposto-que (S ?) because the raising construction is-supposedto translates to a bridge construction dpressuposto-que and cannot adjoin anywhere in the tree for ~-capaz-de (the translation of beable-to) because there is no S-labeled adjunction site.", "cite_spans": [ { "start": 46, "end": 51, "text": "(S ?)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "The correct target derivation: a:voar ~l:~-capaz-de(VP) ~2:~-pressuposto-que (S) 1The subject is omitted to simplify the diagram. Figure 1 : Sample elementary trees for \"supposed to be able to fly\" which yields the translation in sentence (2),", "cite_spans": [ { "start": 77, "end": 80, "text": "(S)", "ref_id": null } ], "ref_spans": [ { "start": 130, "end": 138, "text": "Figure 1", "ref_id": null } ], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "(2) t~ pressuposto que X 6 capaz de voar.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "is not isomorphic to the source. Worse, this non-isomorphism is unbounded, because the bridge verb pressuposto may have to migrate across any number of intervening raising verbs to find an ancestor that contains an appropriate adjunction site: This sort of non-local non-isomorphic transfer cannot be handled in a synchronous TAG that has an isomorphism restriction on derivation trees\u2022 On the other hand, we do not wish to return to the original non-local formulation of synchronous TAG (Shieber and because the non-local inheritance of links on the derived tree is difficult to implement, and because the non-local formulation can recognize languages beyond the generative power of TAG. Rambow, Wier and Vijay-Shanker themselves introduce D-Tree Grammar (Rambow et al., 1995) and Candito and Kahane introduce the DTG variant Graph Adjunction Grammar (Candito and Kahane, 1998b) in order to solve this problem using a derivation process that mirrors composition more directly, but both involve potentially significantly greater recognition complexity than TAG.", "cite_spans": [ { "start": 756, "end": 777, "text": "(Rambow et al., 1995)", "ref_id": "BIBREF8" }, { "start": 852, "end": 879, "text": "(Candito and Kahane, 1998b)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "Our solution is to retain ISTAG, but move the isomorphism restriction from the derivation structure to the predicate-argument attachment structure described in (Joshi and Vijay-Shanker, 1999) . This structure represents the composition of semantic predicates for lexicalized elementary trees, each of which contains a 'predicate' variable associated with the situation or entity that the predicate introduces, and a set of 'argument' variables associated with the foot node and substitution sites in the original elementary tree. The predicates are composed by identifying the predicate variable in one predicate with an argument variable in another, so that the two variables refer to the same situation or entity.", "cite_spans": [ { "start": 160, "end": 191, "text": "(Joshi and Vijay-Shanker, 1999)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Overview", "sec_num": "2" }, { "text": "Composition proceeds from the bottom up on the derivation tree, with adjuncts traversed in order from the lowest to the highest adjunction site in each elementary tree, in much the same way that a parser produces a derivation. Whenever an initial tree is substituted, its predicate variable is identified in the composed structure with an argument variable of the tree it substitutes into. Whenever an auxiliary tree is adjoined, the predicate variable of the tree it adjoins into is identified in the composed structure with one of its own argument variables. In cases of adjunction, an auxiliary tree's semantics can also specify which variable will become the predicate variable of the composed structure for use in subsequent adjunctions at higher adjunction sites: a modifier auxiliary will return the host tree's original predicate variable, and a predicative auxiliary will return its own predicate variable. 2 Since the traversal must 2See (Schabes and Shieber, 1994) and assume the predicative auxiliary tree/31 :beable-to has a predicate variable s2, representing the situation of something being possible, and an argument variable s3, representing the thing that is possible. If fll is now adjoined into a, the composed structure would have sl identified with s3 (since the situation of flying is the thing that is possible), and s2 as an overall predicate variable, so if another tree later adjoins into this composed structure rooted on a, it will predicate over s2 (the situation that flying is possible) rather than over a's original predicate variable sl (the situation of flying by itself). Note that Joshi and Vijay-Shanker do not require the predicate and modifier distinctions, because they can explicitly specify the fates of any number of predicate variables in a tree's semantic representation. For simplicity, we will limit our discussion to only the two possibilities of predicative and modifier auxiliaries, using one predicate variable per tree. If we represent each such predicate-argument attachment as an arc in a directed graph, we can view the predicate-argument attachment structure of a derivation as a dependency graph, in much the same way as Candito and Kahane interpret the original derivation trees (Candito and Kahane, 1998a) . More importantly, we can see that this definition predicts the predicateargument dependencies for sentences (1) and 2 ", "cite_spans": [ { "start": 948, "end": 975, "text": "(Schabes and Shieber, 1994)", "ref_id": "BIBREF9" }, { "start": 2238, "end": 2265, "text": "(Candito and Kahane, 1998a)", "ref_id": "BIBREF1" } ], "ref_spans": [], "eq_spans": [], "section": "Overview", "sec_num": "2" }, { "text": "and Schabes (Shieber and Schabes, 1990) using Synchronous TAG, in that the former preserves the scope ordering of predicative adjunctions, which may be permuted in the latter, altering the meaning of the sentence. 3 It is precisely this scope-preserving property we hope to exploit in our formulation of a dependency-based isomorphic synchronous TAG in the next two sections. However, as Joshi and Vijay-Shanker suggest, the proper treatment of synchronous translation to logical form may require a multicomponent Synchronous TAG analysis in order to handle quantifiers, which is beyond the scope of this paper. For this reason, we will focus on examples in machine translation.", "cite_spans": [ { "start": 4, "end": 39, "text": "Schabes (Shieber and Schabes, 1990)", "ref_id": "BIBREF11" } ], "ref_spans": [], "eq_spans": [], "section": "9{)", "sec_num": null }, { "text": "Obtaining Source Dependencies", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "If we assume that this attachment structure captures a sentence's semantic dependencies, then in order to preserve semantic dependencies in synchronous TAG translation, we will need to obtain this structure from a source derivation and then construct a target derivation with an isomorphic structure.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "The first algorithm we present obtains semantic dependencies for derivations by keeping track of an additional field in each chart item during parsing, corresponding to the predicate variable from Section 2. Other than the additional field, the algorithm remains essentially the same as the parsing algorithm described in (Schabes and Shieber, 1994) , so it can be applied as a transducer during recognition, or as a post-process on a derivation forest (Vijay-Shanker and Weir, 1993). Once the desired dependencies are obtained, the forest may be filtered to select a single most-preferred tree using statistics or rule-based selectional restrictions on those dependencies. 4", "cite_spans": [ { "start": 322, "end": 349, "text": "(Schabes and Shieber, 1994)", "ref_id": "BIBREF9" } ], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "For calculating dependencies, we define a function arg(~) to return the argument position associated with a substitution site or foot node ~? in elementary tree V. Let a dependency be defined as a labeled arc (\u00a2, l, ~b), from predicate \u00a2 to predicate \u00a2 with label I.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "\u2022 For each tree selected by \u00a2, set the predicate variable of each anchor item to \u00a2.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "3See (Joshi and Vijay-Shanker, 1999 ) for a complete description.", "cite_spans": [ { "start": 5, "end": 35, "text": "(Joshi and Vijay-Shanker, 1999", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "4See (Schuler, 1998) for a discussion of statistically filtering TAG forests using semantic dependencies.", "cite_spans": [ { "start": 5, "end": 20, "text": "(Schuler, 1998)", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "\u2022 For each substitution of initial tree a\u00a2 with predicate variable w into \"),\u00a2 at node address U, emit (\u00a2, arg(v , r/), w)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "\u2022 For each modifier adjunction of auxiliary tree/3\u00a2 into tree V\u00a2 with predicate variable X, emit (\u00a2, arg(p, FOOT), X) and set the predicate variable of the composed item to X.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "\u2022 For each predicative adjunction of auxiliary tree /3\u00a2 with predicate variable w into tree \"),\u00a2 with predicate variable X, emit (\u00a2, arg(/3, FOOT), X) and set the predicate variable of the composed item to w.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "\u2022 For all other productions, propagate the predicate variable up along the path from the main anchor to the root.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "Since the number of possible values for the additional predicate variable field is bounded by n, where n is the number of lexical items in the input sentence, and none of the productions combine more than one predicate variable, the complexity of the dependency transducing algorithm is O(nT).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "This algorithm can be applied to the example derivation tree in Section 1, a:fly I /31 :be-able-to(VP) I /32 :is-supposed-to(VP) which resembles the stacked derivation tree for Candito and Kahane's example 5a, \"Paul claims Mary said Peter left.\" First, we adjoin/32 :is-supposed-to at node VP of/31 :be-able-to, which produces the dependency (is-supposed-to,0,be-able-to}. Then we adjoin ~31:be-able-to at node VP of a:fly, which produces the dependency (be-able-to,0,fly). The resulting dependencies are represented graphi-Cally in the dependency structure below: \u00a20 :supposed-to This example is relatively straightforward, simply reversing the direction of adjunction dependencies as described in (Candito and Kahane, 1998a) , but this algorithm can transduce the correct isomorphic dependency structure for the Portuguese derivation as well, similar to the distributed derivation tree in Candito and Kahane's example 5b, \"Paul claims Mary seems to adore hot dogs,\" (Rambow et al., 1995) , where there is no edge corresponding to the dependency between the raising and bridge verbs: We begin by adjoining ~1 :g-capaz-de at node VP of c~:voar, which produces the dependency (~-capaz-de, 0,voar), just as before. Then we adjoin p2:~-pressuposto-que at node S of c~:voar. This time, however, we must observe the predicate variable of the chart item for c~:voar which was updated in the previous adjunction, and now references ~-capaz-de instead of voar. Because the transduction rule for adjunction uses the predicate variable of the parent instead of just the predicate, the dependency produced by the adjunetion of ~2 is (~-pressuposto-que, 0,~capaz-de), yielding the graph:", "cite_spans": [ { "start": 699, "end": 726, "text": "(Candito and Kahane, 1998a)", "ref_id": "BIBREF1" }, { "start": 968, "end": 989, "text": "(Rambow et al., 1995)", "ref_id": "BIBREF8" } ], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "As Candito and Kahane point out, this derivation tree does not match the dependency structure of the sentence as described in Meaning Text Theory (Mel'cuk, 1988) , because there is no edge in the derivation corresponding to the dependency between surprise and have-to (the necessity of Paul's staying is what surprises Mary, not his staying in itself). Using the above algorithm, however, we can still produce the desired dependency structure: The derivation examples above only address the preservation of dependencies through adjunction.", "cite_spans": [ { "start": 146, "end": 161, "text": "(Mel'cuk, 1988)", "ref_id": "BIBREF6" } ], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "Let us now attempt to preserve both substitution and adjunction dependencies in transducing a sentence based on Candito and Kahane's example 5c, \"That Paul has to stay surprised Mary,\" in order to demonstrate how they interact. 5 We begin with the derivation tree: al :surprise c~2 :stay(S0) c~4 :Mary(NPl) c~a:Paul(NP0) ~:have-to(VP)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "5We have replaced want to in the original example with have to in order to highlight the dependency structure and set aside any translation issues related to PRO control.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3", "sec_num": null }, { "text": "Once a source derivation is selected from the parse forest, the predicate-argument dependencies can be read off from the items in the forest that constitute the selected derivation. The resulting dependency graph can then be mapped to a forest of target derivations, where each predicate node in the source dependency graph is linked to a set of possible elementary trees in the target grammar, each of which is instantiated with substitution or adjunction edges leading to other linked sets in the forest. The elementary trees in the target forest are determined by the predicate pairs in the transfer lexicon, and by the elementary trees that can realize the translated targets. The substitution and adjunction edges in the target forest are determined by the argument links in the transfer lexicon, and by the substitution and adjunction configurations that can realize the translated targets' dependencies.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "Mapping dependencies into substitutions is relatively straightforward, but we have seen in Section 2 that different adjunction configurations (such as the raising and bridge verb ad-junctions in sentences (1) and (2)) can correspond to the same dependency graph, so we should expect that some dependencies in our target graph may correspond to more than one adjunction configuration in the target derivation tree. Since a dependency may be realized by adjunctions at up to n different sites, an unconstrained algorithm would require exponential time to find a target derivation in the worst case. In order to reduce this complexity, we present a dynamic programming algorithm for constructing a target derivation forest in time proportional to O(n 4) which relies on a restriction that the target derivations must preserve the relative scope ordering of the predicates in the source dependency graph.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "This restriction carries the linguistic implication that the scope ordering of adjuncts is part of the meaning of a sentence and should not be re-arranged in translation. Since we exploit a notion of locality similar to that of Isomorphic Synchronous TAG, we should not expect the generative power of our definition to exceed the generative power of TAG, as well.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "First, we define an ordering of predicates on the source dependency graph corresponding to a depth-first traversal of the graph, originating at the predicate variable of the root of the source derivation, and visiting arguments and modifiers in order from lowest to highest scope. In other words, arguments and modifiers will be ordered from the bottom up on the elementary tree structure of the parent, such that the foot node argument of an elementary tree has the lowest scope among the arguments, and the first adjunct on the main (trunk) anchor has the lowest scope among the modifiers.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "Arguments, which can safely be permuted in translation because their number is finitely bounded, are traversed entirely before the parent; and modifiers, which should not be permuted because they may be arbitrarily numerous, are traversed entirely after the parent. This enumeration will roughly correspond to the scoping order for the adjuncts in the source derivation, while preventing substituted trees from interrupting possible scoping configurations. We can now identify all the descendants of any elementary tree in a derivation because they will form a consecutive series in the enumeration described above. It therefore provides a convenient way to generate a target derivation forest that preserves the scoping information in the source, by 'parsing' the scope-ordered string of elementary trees, using indices on this enumeration instead of on a string yield.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "It is important to note that in defining this algorithm, we assume that all trees associated with a particular predicate will use the same argument structure as that predicate. 6 We also assume that the set of trees associated with a particular predicate may be filtered by transferring information such as mood and voice from source to target predicates.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "Apart from the different use of indices, the algorithm we describe is exactly the reverse of the transducer described in Section 3, taking a dependency graph 79 and producing a TAG derivation forest containing exactly the set of derivation trees for which those dependencies hold. Here, as in a parsing algorithm, we define forest items as tuples of (~/\u00a2, 'q, _1_, i,j, X) where a, ~, and 7 are elementary trees with node'O, \u00a2 and \u00a2 are predicates, X and w be predicate variables, and T and _1_ are delimiters tbr opening and closing adjunction, but now let i, j, and k refer to the indices on the scoping enumeration described above, instead of on an input string. In order to reconcile scoping ranges for substitution, we must also define a function first (C) to return the leftmost (lowest) edge of the \u00a2's range in the scope enumeration, and last(C) to return the rightmost (highest) edge of the \u00a2's range in the scope enumeration.", "cite_spans": [ { "start": 350, "end": 372, "text": "(~/\u00a2, 'q, _1_, i,j, X)", "ref_id": null }, { "start": 758, "end": 761, "text": "(C)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "\u2022 For each tree 7 mapped from predicate \u00a2 at scope i, introduce (~,\u00a2, first(C), i + 1, \u00a2}.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "\u2022 If (\u00a2,arg(7,~) ,co) E 79, try substitution of c~ into 3':", "cite_spans": [ { "start": 5, "end": 16, "text": "(\u00a2,arg(7,~)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "(c~\u00a2, ROOT, T, first(co), last(co), co)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Obtaining Target Derivations", "sec_num": "4" }, { "text": "\u2022 If (\u00a2, arg(/3, FOOT), X) E 79, try modifier adjunction of fl into -),: (V~,~h_l_,i,j,x) (/3\u00a2, ROOT, T, j, k, w) (V\u00a2, ~, -l-, i, k, x) \u2022 If (\u00a2, arg(/3, FOOT), X) E 79, try predicative adjunction of/3 into V: (V\u00a2, ~, _I_, i, j, x) (/3\u00a2, ROOT, T, j, k, w) (V\u00a2, ~, T, i, k, w) \u2022 Apply productions for nonterminal projection as in the transducer algorithm, propagating index ranges and predicative variables up along the path from the main anchor to the root.", "cite_spans": [ { "start": 73, "end": 89, "text": "(V~,~h_l_,i,j,x)", "ref_id": null }, { "start": 90, "end": 95, "text": "(/3\u00a2,", "ref_id": null }, { "start": 96, "end": 101, "text": "ROOT,", "ref_id": null }, { "start": 102, "end": 104, "text": "T,", "ref_id": null }, { "start": 105, "end": 107, "text": "j,", "ref_id": null }, { "start": 108, "end": 110, "text": "k,", "ref_id": null }, { "start": 111, "end": 113, "text": "w)", "ref_id": null }, { "start": 114, "end": 135, "text": "(V\u00a2, ~, -l-, i, k, x)", "ref_id": null }, { "start": 209, "end": 213, "text": "(V\u00a2,", "ref_id": null }, { "start": 214, "end": 216, "text": "~,", "ref_id": null }, { "start": 217, "end": 221, "text": "_I_,", "ref_id": null }, { "start": 222, "end": 224, "text": "i,", "ref_id": null }, { "start": 225, "end": 227, "text": "j,", "ref_id": null }, { "start": 228, "end": 236, "text": "x) (/3\u00a2,", "ref_id": null }, { "start": 237, "end": 242, "text": "ROOT,", "ref_id": null }, { "start": 243, "end": 245, "text": "T,", "ref_id": null }, { "start": 246, "end": 248, "text": "j,", "ref_id": null }, { "start": 249, "end": 251, "text": "k,", "ref_id": null }, { "start": 252, "end": 259, "text": "w) (V\u00a2,", "ref_id": null }, { "start": 260, "end": 262, "text": "~,", "ref_id": null }, { "start": 263, "end": 265, "text": "T,", "ref_id": null }, { "start": 266, "end": 268, "text": "i,", "ref_id": null }, { "start": 269, "end": 271, "text": "k,", "ref_id": null }, { "start": 272, "end": 274, "text": "w)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "7, \u00b1, , ,-)", "sec_num": null }, { "text": "Since none of the productions combine more than three indices and one predicate variable, and since the indices and predicate variable may have no more than n distinct values, the algorithm runs in O(n 4) time. Note that one of the indices may be redundant with the predicate variable, so a more efficient implementation might be possible in dO(n3).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "7, \u00b1, , ,-)", "sec_num": null }, { "text": "We can demonstrate this algorithm by translating the English dependency graph from Section 1 into a derivation tree for Portuguese. First, we enumerate the predicates with their relative scoping positions: at the bottom, we assign to these constituents the relative scoping ranges of 1-2, 2-3, and 3-$, respectively, where $ is a terminal symbol.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "7, \u00b1, , ,-)", "sec_num": null }, { "text": "There is also a dependency from is-supposedto to be-able-to allowing us to adjoin /32:dpressuposto-que to /31:d-capaz-de to make it cover the range from 2 to $, but there would be no S node to host its adjunction, so this possibility can not be added to the forest. We can, however, adjoin/32:d-pressuposto-que to the instance of a:voar extending to/31 :d-capaz-de that covers the range from 1 to 3, resulting in a complete analysis of the entire scope from 1 to $, (from (~:voar to/32:pressuposto) rooted on voar: (O~voar, l,2,..) (/3capaz, 2, 3, ..) (/3press, 3, $, ..) (/3capaz,2,3,..> (/3vai,3,$,..> (avoar, l,2,..) (/3capaz,2,3,..> (/3press, 3,$,..> Since there is a dependency from be-able-to to fly, we can adjoin/31:d-capaz-de to a:voar such that it covers the range of scopes from 1 to 3 (from roar to d-capaz-de), so we add this possibility to the forest.", "cite_spans": [ { "start": 278, "end": 341, "text": "(avoar, l,2,..> (/3capaz,2,3,..> (/3vai,3,$,..> (avoar, l,2,..)", "ref_id": null }, { "start": 342, "end": 376, "text": "(/3capaz,2,3,..> (/3press, 3,$,..>", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "Although we can still adjoin/31 :ser-capaz-de at the VP node of a:voar, we will have nowhere to adjoin /32:vai, since the VP node of a:voar is now occupied, and only one predicative tree may adjoin at any node. 7", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "(avoar, 1, 2,..) (t3capaz, 2, 3, ..) (/3vai, 3, $, ..) (avoar, 1, 3, capaz> (avoar , l, 2, ..) (/3capaz, 2, 3, -.) (/3;ress, 3,$,..) (avoar, 1, 3, capaz)", "cite_spans": [ { "start": 55, "end": 84, "text": "(avoar, 1, 3, capaz> (avoar ,", "ref_id": null } ], "ref_spans": [ { "start": 85, "end": 132, "text": "l, 2, ..) (/3capaz, 2, 3, -.) (/3;ress, 3,$,..)", "ref_id": "FIGREF1" } ], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "7See (Schabes and Shieber, 1994) for the motivations of this restriction.", "cite_spans": [ { "start": 5, "end": 32, "text": "(Schabes and Shieber, 1994)", "ref_id": "BIBREF9" } ], "ref_spans": [], "eq_spans": [], "section": "I", "sec_num": null }, { "text": "Fortunately, we can also realize the dependency between vai and ser-capaz-de by adjoining/32 :vai at the VP.