| { | |
| "paper_id": "S12-1022", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T15:23:36.961868Z" | |
| }, | |
| "title": "Expanding the Range of Tractable Scope-Underspecified Semantic Representations", | |
| "authors": [ | |
| { | |
| "first": "Mehdi", | |
| "middle": [], | |
| "last": "Manshadi", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Rochester Rochester", | |
| "location": { | |
| "postCode": "14627", | |
| "region": "NY" | |
| } | |
| }, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "James", | |
| "middle": [], | |
| "last": "Allen", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Rochester Rochester", | |
| "location": { | |
| "postCode": "14627", | |
| "region": "NY" | |
| } | |
| }, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "Over the past decade, several underspecification frameworks have been proposed that efficiently solve a big subset of scopeunderspecified semantic representations within the realm of the most popular constraint-based formalisms. However, there exists a family of coherent natural language sentences whose underspecified representation does not belong to this subset. It has remained an open question whether there exists a tractable superset of these frameworks, covering this family. In this paper, we show that the answer to this question is yes. We define a superset of the previous frameworks, which is solvable by similar algorithms with the same time and space complexity.", | |
| "pdf_parse": { | |
| "paper_id": "S12-1022", | |
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| "abstract": [ | |
| { | |
| "text": "Over the past decade, several underspecification frameworks have been proposed that efficiently solve a big subset of scopeunderspecified semantic representations within the realm of the most popular constraint-based formalisms. However, there exists a family of coherent natural language sentences whose underspecified representation does not belong to this subset. It has remained an open question whether there exists a tractable superset of these frameworks, covering this family. In this paper, we show that the answer to this question is yes. We define a superset of the previous frameworks, which is solvable by similar algorithms with the same time and space complexity.", | |
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| "section": "Abstract", | |
| "sec_num": null | |
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| "body_text": [ | |
| { | |
| "text": "Scope ambiguity is a major source of ambiguity in semantic representation. For example, the sentence 1. Every politician has a website. has at least two possible interpretations, one in which each politician may have a different website (i.e., Every has wide scope) and one in which there is a unique website for all the politicians (i.e., Every has narrow scope). Since finding the most preferred reading automatically is very hard, the most widely adopted solution is to use an Underspecified Representation (UR), that is to encode the ambiguity in the semantic representation and leave scoping underspecified.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "In an early effort, Woods (1986) developed an unscoped logical form where the above sentence is represented (roughly) as the formula:", | |
| "cite_spans": [ | |
| { | |
| "start": 20, | |
| "end": 32, | |
| "text": "Woods (1986)", | |
| "ref_id": "BIBREF19" | |
| } | |
| ], | |
| "ref_spans": [], | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "To obtain a fully scoped formula, the quantifiers are pulled out one by one and wrapped around the formula. If we pull out Every first, we produce the fully-scoped formula:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Has( Every x P olitician , A y W ebsite )", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "3. A(y, W ebsite(y), Every(x, P olitician(x), Has(x, y))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Has( Every x P olitician , A y W ebsite )", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "If we had pulled out A first, we would have had the other reading, with Every having wide scope. Hobbs and Shieber (1987) extend this formalism to support operators (such as not) and present an enumeration algorithm that is more efficient than the naive wrapping approach.", | |
| "cite_spans": [ | |
| { | |
| "start": 97, | |
| "end": 121, | |
| "text": "Hobbs and Shieber (1987)", | |
| "ref_id": "BIBREF9" | |
| } | |
| ], | |
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| "section": "Has( Every x P olitician , A y W ebsite )", | |
| "sec_num": "2." | |
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| "text": "Since the introduction of Quasi Logical Form (Alshawi and Crouch, 1992) , there has been a lot of work on designing constraint-based underspecification formalisms where the readings of a UR are not defined in a constructive fashion as shown above, but rather by a set of constraints. A fully-scoped structure is a reading iff it satisfies all the constraints. The advantage of these frameworks is that as the processing goes deeper, new (say pragmatically-driven) constraints can be added to the representation in order to filter out unwanted readings. Hole Semantics (Bos, 1996; Bos, 2002) , Constraint Language for Lambda Structures (CLLS) (Egg et al., 2001) , and Minimal Recursion Semantics (MRS) (Copestake et al., 2001) are among these frameworks.", | |
| "cite_spans": [ | |
| { | |
| "start": 45, | |
| "end": 71, | |
| "text": "(Alshawi and Crouch, 1992)", | |
| "ref_id": "BIBREF0" | |
| }, | |
| { | |
| "start": 568, | |
| "end": 579, | |
| "text": "(Bos, 1996;", | |
| "ref_id": "BIBREF3" | |
| }, | |
| { | |
| "start": 580, | |
| "end": 590, | |
| "text": "Bos, 2002)", | |
| "ref_id": "BIBREF4" | |
| }, | |
| { | |
| "start": 642, | |
| "end": 660, | |
| "text": "(Egg et al., 2001)", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 701, | |
| "end": 725, | |
| "text": "(Copestake et al., 2001)", | |
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| ], | |
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| "section": "Has( Every x P olitician , A y W ebsite )", | |
| "sec_num": "2." | |
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| "text": "In an effort to bridge the gap between the above formalisms, a graph theoretic model of scope underspecification was defined by Bodirsky et al. (2004) , called Weakly Normal Dominance Graphs. This framework and its ancestor, Dominance Constraints (Althaus et al., 2003) , are broad frameworks for solving constrained tree structures in general. When it comes to scope underspecification, some of the terminology becomes counter-intuitive. Therefore, here we first define (scope) Underspecification Graphs (UG), a notational variant of weakly normal dominance graphs, solely defined to model scope underspecification. 1 Figure 1 shows a UG for the following sentence.", | |
| "cite_spans": [ | |
| { | |
| "start": 128, | |
| "end": 150, | |
| "text": "Bodirsky et al. (2004)", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 247, | |
| "end": 269, | |
| "text": "(Althaus et al., 2003)", | |
| "ref_id": "BIBREF1" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 619, | |
| "end": 627, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Has( Every x P olitician , A y W ebsite )", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "The big circles and the dot nodes are usually referred to as the hole nodes (or simply holes) and the label nodes (or simply labels) respectively. The left and the right holes of each quantifier are placeholders for the restriction and the body of the quantifier. A fully scoped structure is built by plugging labels into holes, as shown in Figure 2 (a). The dotted edges represent the constraints. For example, the constraint from the restriction hole of Every(x) to the node Politician(x) states that this label node must be within the scope of the restriction of Every(x) in every reading of the sentence. The constraint edge from Every(x) to Run(x) forces the binding constraint for variable x; that is variable x in Run(x) must be within the scope of its quantifier. Figure 2 (b) represents the other possible reading of the sentence. Now consider the sentence:", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 341, | |
| "end": 349, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| }, | |
| { | |
| "start": 772, | |
| "end": 780, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "5. Every politician, whom I know a child of, probably runs.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "with its UG shown in Figure 3 . This sentence contains a scopal adverbial (a.k.a. fixed-scopal; cf. Copestake et al. (2005) ), the word Probably. Since in general, quantifiers can move inside or outside a scopal operator, the scope of Probably is left underspecified, and hence represented by a hole. It is easy to verify that the corresponding UG has five possible readings, two of which are shown in Figure 4 . There are at least two major algorithmic problems that need to be solved for any given UG U : the satisfiability problem; that is whether there exists any reading satisfying all the constraints in U , and the enumeration problem; that is enumerating all the possible readings of a satisfiable U . Unfortunately, both problems are NP-complete for UG in its general form (Althaus et al., 2003) . This proves that Hole Semantics and Minimal Recursion Semantics are also intractable in their general form (Thater, 2007) . In the last decade, there has been a series of interesting work on finding a tractable subset of those frameworks, broad enough to cover most structures occurring in practice. Those efforts resulted in two closely related tractable frameworks: (dominance) net and weak (dominance) net. Intuitively, the net condition requires the following property. Given a UG U , for every label node in U with n holes, if the node together with all its holes is removed from U , the remaining part is composed of at most n (weakly) connected components. A difference between net and weak net is that in nets, label- to-label constraints (e.g. the constraint between Every(x) and Run(x) in Figure 1 ) are not allowed.", | |
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| { | |
| "start": 100, | |
| "end": 123, | |
| "text": "Copestake et al. (2005)", | |
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| "end": 804, | |
| "text": "(Althaus et al., 2003)", | |
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| "end": 928, | |
| "text": "(Thater, 2007)", | |
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| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 21, | |
| "end": 29, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 402, | |
| "end": 410, | |
| "text": "Figure 4", | |
| "ref_id": "FIGREF3" | |
| }, | |
| { | |
| "start": 1606, | |
| "end": 1614, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
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| "text": "Using a sample grammar for CLLS, conjecture that the syntax/semantics interface of CLLS only generates underspecified representations that follow the definition of net and hence can be solved in polynomial time. They also prove that the same efficient algorithms can be used to solve the underspecification structures of Hole Semantics which satisfy the net condition.", | |
| "cite_spans": [], | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "Unlike Hole Semantics and CLLS, MRS implicitly carries label-to-label constraints; hence the concept of net could not be applied to MRS. In order to address this, define the notion of weak net and conjecture that it covers all semantically complete MRS structures occurring in practice. Fuchss et al. (2004) supported the claim by investigating MRS structures in the Redwoods corpus (Oepen et al., 2002) . Later coherent sentences were found in other corpora or suggested by other researchers (see Section 6.2.2 in Thater (2007)), whose UR violates the net condition, invalidating the conjecture. However, violating the net condition occurs in a similar way in those examples, suggesting a family of non-net structures, characterized in Section 4.2. Since then, it has been an open question whether there exists a tractable superset of weak nets, covering this family of non-net UGs.", | |
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| { | |
| "start": 287, | |
| "end": 307, | |
| "text": "Fuchss et al. (2004)", | |
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| }, | |
| { | |
| "start": 383, | |
| "end": 403, | |
| "text": "(Oepen et al., 2002)", | |
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| } | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
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| { | |
| "text": "In the rest of this paper, we answer this question. We modify the definition of weak net to define a superset of it, which we call super net. Super net covers the above mentioned family of non-net structures, yet is solvable by (almost) the same algorithms as those solving weak nets with the same time and space complexity.", | |
| "cite_spans": [], | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
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| "text": "The structure of the paper is as follows. We define our framework in Section 2 and present the polynomial-time algorithms for its satisfiability and enumeration problems in Section 3. In Section 4, we compare our framework with nets and weak nets. Section 5 discusses the related work, and Section 6 summarizes this work and discusses future work.", | |
| "cite_spans": [], | |
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| "section": "Every child of a politician runs.", | |
| "sec_num": "4." | |
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| { | |
| "text": "We first give a formal definition of underspecification graph (UG). We then define super net as a subset of UG. In the following definitions, we openly borrow the terminology from Hole Semantics, Dominance Constraints, and MRS, in order to avoid inventing new terms to name old concepts.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Super net", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Definition 1 (Fragments). Consider L a set of labels, H a set of holes, and S a set of directed solid edges from labels to holes, such that F = (L H, S) is a forest of ordered trees of depth at most 1, whose root and only the root is a label node. Each of these trees is called a fragment.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Super net", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Following this definition, the number of trees in F (including single-node trees) equals the number of labels. For example, if we remove all the dotted edges in Figure 1 , we obtain a forest of 5 fragments.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 161, | |
| "end": 169, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Super net", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Definition 2 (Underspecification Graph). Let F = (L H, S) be a forest of fragments and C be a set of directed dotted edges from L H to L, called the set of constraints. 2 U = (L H, S C) is called an underspecification graph or UG.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Super net", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Definition 3 (Plugging). (Bos, 1996 ) Given a UG U = (L H, S C), a plugging P is a total one-to-one function from H to L.", | |
| "cite_spans": [ | |
| { | |
| "start": 25, | |
| "end": 35, | |
| "text": "(Bos, 1996", | |
| "ref_id": "BIBREF3" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In Figure 1 , if l A , l E , l P , l C , and l R represent the nodes labeled by A(y), Every(x), Politician(y), Child(x,y), and Run(x) respectively and h r A (h b A ) and h r E (h b E ) represent the restriction (body) hole of A and Every respectively, then P in (6) is a plugging.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 3, | |
| "end": 11, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "6. P = {(h r A , l P ), (h b A , l C ), (h r E , l A ), (h b E , l R )}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We use T U,P to refer to the graph, formed from U by removing all the constraints and plugging P (h) into h for every hole h. For example if U is the UG in Figure 1 and P is the plugging in (6), then T U,P is the graph shown in Figure 2 (a).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 156, | |
| "end": 164, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 228, | |
| "end": 236, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
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| "text": "Definition 4 (Permissibility/Solution). T U,P satisfies the constraint (u,v) in U , iff u dominates 3 v in T U,P . 4 A plugging P is permissible, iff T U,P is a forest satisfying all the constraints in U . T U,P is called a solution of U iff P is a permissible plugging. In informal contexts, solutions are sometimes referred to as readings.", | |
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| "section": "Figures 1 and 3 each represent a UG.", | |
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| "text": "It is easy to see that the plugging in (6) is a permissible plugging for the UG in Figure 1 , and hence Figure 2 Throughout the rest of this paper, unless otherwise specified, UGs are assumed to be weakly connected, hence solutions are tree structures. 6 Lemma 2. (Bodirsky et al., 2004 ) Given a UG U and a solution T of U , if the nodes u and v in U are connected using an undirected path p, there exists a node w on p such that w dominates both u and v in T.", | |
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| "end": 286, | |
| "text": "(Bodirsky et al., 2004", | |
| "ref_id": "BIBREF2" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 83, | |
| "end": 91, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 104, | |
| "end": 112, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| } | |
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| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
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| "text": "This Lemma is proved using induction on the length of p. As mentioned before, satisfiability and enumeration are two fundamental problems to be solved for a UG. A straightforward approach is depicted in Figure 5 . We pick a label l; remove it from U ; recursively solve each of the resulting weakly connected components (WCCs; cf. footnote 2) and plug the root of the returned trees into the corresponding holes of l. A problem to be addressed though is whether there exists any solution rooted at l. This leads us to the following definition. Definition 5 (Freeness). (Bodirsky et al., 2004 ) A label l in U is called a free node, iff there exists some solution of U rooted at l. The fragment rooted at l is called a free fragment.", | |
| "cite_spans": [ | |
| { | |
| "start": 569, | |
| "end": 591, | |
| "text": "(Bodirsky et al., 2004", | |
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| } | |
| ], | |
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| { | |
| "start": 203, | |
| "end": 211, | |
| "text": "Figure 5", | |
| "ref_id": "FIGREF5" | |
| } | |
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| "section": "Figures 1 and 3 each represent a UG.", | |
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| "text": "The following proposition states the necessary conditions for a label (or fragment) to be free. 7 Proposition 3. Let l in U be the root of a fragment F with m holes. l is a free node of U , only if P3a. l has no incoming (constraint) edge; P3b. Every distinct hole of F lies in a distinct WCC in U \u2212l; P3c. U \u2212F consists of at least m WCCs. Proof. The first condition is trivial. To see why the second condition must hold, let T be a solution rooted at l, and assume to the contrary that h 1 and h 2 lie in the same WCC in U \u2212 l. From Lemma 2, all the nodes in this WCC must be in the scope of both h 1 and h 2 . But this is not possible, because T is a tree. The third condition is proved similarly. Assume to the contrary that U \u2212F has m \u2212 1 WCCs. From Lemma 2, all the nodes in a WCC must be in the scope of a single hole of F . But there are m holes and only m \u2212 1 WCCs. It means that one of the holes in T is left unplugged. Contradiction! The motivation behind defining super nets is to find a subset of UG for which these conditions are also sufficient. The following concept from Althaus et al. (2003) plays an important role. Definition 6 (Hypernormal Connectedness). Given a UG U , a hypernormal path is an undirected path 8 with no two consecutive constraint edges emanating from the same node. Node u is hypernormally connected to node v iff there is at least one hypernormal path between the two. U is called hypernormally connected iff every pair of nodes in U are hypernormally connected.", | |
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| "section": "Figures 1 and 3 each represent a UG.", | |
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| "text": "For example, in Figure 2 , p 2 is a hypernormal path, but p 1 is not. In spite of that, the whole graph is hypernormally connected. 9 The following simple notion will also come handy. Definition 8 guarantees the following property.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 16, | |
| "end": 24, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Lemma 4. For a super net U and a fragment F of U with m holes, which satisfies the conditions in Proposition 3, U \u2212 F consists of exactly m WCCs, each of which is a super net.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Proof sketch. The detailed proof of this Lemma is long. Therefore, we sketch the proof here and leave the details for a longer paper. First, we show that U \u2212F consists of exactly m WCCs. Following conditions (D8b) and (D8c), no matter what structure F has, U\u2212F consists of at most m WCCs. On the other hand, based on condition (P3c), U\u2212F has at least m WCCs. Therefore, U \u2212F has exactly m WCCs. To prove that each WCC in U \u2212F is a super net, all we need to prove is that if two nodes u and v, which do not belong to F , are hypernormally connected in U , they are also hypernormally connected in U\u2212F . This is proved by showing that there is no hypernormal path between u and v in U that visits some node of F . Suppose that F is an open-hole fragment rooted at l, as in Figure 2 (b) (the two other cases are proved similarly) and assume to the contrary that there is a hypernormal path p between u and v that visits some node of F . One of the following three cases holds.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 771, | |
| "end": 779, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "i. p visits exactly one node of F . ii. p visits (at least) two holes of F . iii. p visits l and exactly one hole of F .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "All the three cases results in a contradiction: (i) proves that p is not hypernormal; (ii) proves that F is not a free fragment because it violates condition (P3b); and (iii) proves that U is not a super net because F violates condition (D8c). Proposition 5. If U is a satisfiable super net, the necessary freeness conditions in Proposition 3 are also sufficient.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Proof sketch. Let F rooted at l be a fragment satisfying the three conditions in Proposition 3. Among all the solutions of U , we pick a solution T in which the depth d of l is minimal. Using proof by contradiction, we show that d = 0, which proves l is the root of T . If d > 0, there is some node u that outscopes l (Figure 8(a) ). Lemma 2 and 4 guarantee that at least one of the trees in Figures 8(b,c) is a solution of U . So U has a solution in which, the depth of l is smaller than d. Contradiction!", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 318, | |
| "end": 330, | |
| "text": "(Figure 8(a)", | |
| "ref_id": "FIGREF9" | |
| }, | |
| { | |
| "start": 392, | |
| "end": 406, | |
| "text": "Figures 8(b,c)", | |
| "ref_id": "FIGREF9" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Figures 1 and 3 each represent a UG.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Following Lemma 4 and Proposition 5, Table 1 gives the algorithms for the satisfiability (SAT), and the enumeration (ENUM) of super nets.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 37, | |
| "end": 44, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "SAT and ENUM algorithms", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "Theorem 6. ENUM and SAT are correct.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "SAT and ENUM algorithms", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "Proof sketch. Using Lemma 4 and induction on the depth of the recursion, it is easy to see that if ENUM or SAT returns a tree T , T is a solution of U . This proves that ENUM and SAT are sound. An inductive proof is used to prove the completeness as well. Consider a solution T of depth n of U ( Figure 5 ). It can be shown that T 1 and T 2 must be the solutions to U 1 and U 2 . Therefore based on the induction assumption they are generated by Solve(G), hence T is also generated by Solve(G).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 296, | |
| "end": 304, | |
| "text": "Figure 5", | |
| "ref_id": "FIGREF5" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "SAT and ENUM algorithms", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "Let U = (L H, S C). The running time of the algorithms depends on the depth of the recursion which is equal to the number of fragments/labels, |L|. At each depth it takes O(|U |) to find the set of free fragments (Bodirsky et al., 2004) and also to compute U \u2212 F for some free fragment F .", | |
| "cite_spans": [ | |
| { | |
| "start": 213, | |
| "end": 236, | |
| "text": "(Bodirsky et al., 2004)", | |
| "ref_id": "BIBREF2" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "SAT and ENUM algorithms", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "1. If U contains a single (label) node, return U . 2. Pick a free fragment F with m holes rooted at l, otherwise fail. // For SAT: pick arbitrarily. // For ENUM: pick non-deterministically. |) step. Therefore the worst-case time complexity of SAT and each branch of ENUM is quadratic in the size of U .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 190, | |
| "end": 192, | |
| "text": "|)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Solve(U)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "3. Let U 1 , U 2 , \u2022 \u2022 \u2022 , U m be WCCs of U \u2212F . 4. Let T i = Solve(U i ) for i = 1 \u2022 \u2022 \u2022 m.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Solve(U)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Although net is a subset of weak net, to better understand the three frameworks, we first define net.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Super net versus weak net", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Net was first defined by , in order to find a subset of Hole Semantics that can be solved in polynomial-time. Nets do not contain any label-to-label constraints. In fact, out of the three possible structures that super net allows for a fragment F (Definition 9), net only allows for the first one, that is open-root.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Net", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Definition 10 (Net). (Thater, 2007 ) Let U be a UG with no label-to-label constraints. U is called a net iff for every fragment F :", | |
| "cite_spans": [ | |
| { | |
| "start": 21, | |
| "end": 34, | |
| "text": "(Thater, 2007", | |
| "ref_id": "BIBREF18" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Net", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "D10a. F has no open hole. D10b. If l 1 , l 2 are two dominance children of a hole h of F , then l 1 and l 2 are hypernormally connected in U \u2212h.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Net", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "The root of F is open, therefore (D8a) subsumes (D10a). Condition (D10b) is exactly the same as (D8b). Therefore, super net is a superset of net. Strictness of the superset relationship is trivial.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Net", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Weak net was first introduced by , in order to find a tractable subset of MRS. In order to model MRS, weak net allows for label-tolabel constraints, but to stay a tractable framework it forces the following restrictions.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Definition 11 (Weak net). (Thater, 2007 ) A UG U is a weak net iff for every fragment F :", | |
| "cite_spans": [ | |
| { | |
| "start": 26, | |
| "end": 39, | |
| "text": "(Thater, 2007", | |
| "ref_id": "BIBREF18" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "D11a. F has exactly one open node.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "D11b. If l 1 , l 2 are two dominance children of a node u of F , then l 1 and l 2 are hypernormally connected in U \u2212u.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Weak nets suffer from two limitations with respect to super nets.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "First, out of the three possible types of fragment allowed by super net (Definition 9), weak net only allows for the first two; open-root and openhole. In practice this becomes an issue only if new constraints are to be added to a UG after syntax/semantic interface. Since weak net requires one node of every fragment to be open, a constraint cannot be added if it violates this condition. 11 Second, open-hole fragments in weak nets are more restricted than open-hole fragments in super nets. This is the Achilles' heel of weak nets (D11b). To see why, consider the UG in Figure 3 for the sentence Every politician, whom I know a child of, runs which we presented in Section 1. If F is the fragment for the quantifier Every and l is the root of F , the two dominance children of l are not (hypernormally) connected in U \u2212 l. Therefore, U is not a weak net. All the non-net examples we have found so far behave similarly. That is, there is a quantifier with more than one outgoing dominance edge. Once you remove the quantifier node, the dominance children are no longer weakly (and hence hypernormally) connected, violating condition (D11b). In super net, however, we define case 2 of condition (D8c) such that it does not force dominance children of l to be (hypernormally) connected, allowing for non-net structures such as the one in Figure 3 . 12 Proposition 7. Weak net is a strict subset of super net.", | |
| "cite_spans": [ | |
| { | |
| "start": 390, | |
| "end": 392, | |
| "text": "11", | |
| "ref_id": null | |
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| { | |
| "start": 1349, | |
| "end": 1351, | |
| "text": "12", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 573, | |
| "end": 581, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 1338, | |
| "end": 1346, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Proof. Consider an arbitrary weak net U , and let F be an arbitrary fragment of U rooted at l. Based on condition (D11b) every two dominance children of l are hypernormally connected, so (D8c) holds in this case too.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Therefore, every fragment F satisfies all the conditions in Definition 8, hence U is a super net. This and the fact that Figure 3 is a super net but not a weak net complete the proof.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 121, | |
| "end": 129, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Weak net", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Dominance graphs and their ancestor, dominance constraints, are designed for solving constrained tree structures in general. Therefore, some of the terminology of dominance graph may seem counterintuitive when dealing with scope underspecification. For example the notion of solution in that formalism is broader than what is known as solution in scope underspecification formalisms. As defined there (but translated into our terminology), a solution may contain unplugged holes, or holes plugged with more than one label. This broad notion of solution is computationally less expensive such that an algorithm very similar to the one in Table 1 can be used to solve every weakly normal dominance graph (Bodirsky et al., 2004) . Solution, as defined in this paper (Definition 4), corresponds to the notion of simple leaf-labeled solved forms (a.k.a. configuration) in dominance graphs. Although solutions of a weakly normal dominance graph can be found in polynomial time, finding configurations is NPcomplete. Solvability of underspecification graphs is equivalent to configurability of weakly normal dominance graphs, and hence NP-complete.", | |
| "cite_spans": [ | |
| { | |
| "start": 702, | |
| "end": 725, | |
| "text": "(Bodirsky et al., 2004)", | |
| "ref_id": "BIBREF2" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 637, | |
| "end": 644, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Underspecification graph vs. weakly normal dominance graph", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "We already compared our model with nets and weak nets. Koller and Thater (2007) present another extension of weak nets, downward connected nets. They show that if a dominance graph has a subgraph which is a weak net, it can be solved in polynomial time. This addresses the first limitation of weak nets, discussed in Section 4.2, but it does not solve the second one, because the graph in Figure 3 neither is a weak net, nor has a weak-net subgraph. Downward connected dominance graph, in its general form, goes beyond weakly normal dominance graph (and hence UG), incorporating label-tohole constraints. It remains for future work to investigate whether allowing for label-to-hole constraints adds any value to the framework within the context of scope underspecified semantics, or whether it is possible to model the same effect using hole-to-label and label-to-label constraints. In any case, the same extension can be applied to super nets as well, defining downward connected super nets, a strict super set of downward connected nets, solvable using similar algorithms with the same time/space complexity.", | |
| "cite_spans": [ | |
| { | |
| "start": 55, | |
| "end": 79, | |
| "text": "Koller and Thater (2007)", | |
| "ref_id": "BIBREF10" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 389, | |
| "end": 397, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Related work", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "Another tractable framework presented in the past is our own framework, Canonical Form Underspecified Representation (CF-UR) (Manshadi et al., 2009) , motivated by Minimal Recursion Semantics. CF-UR is defined to characterize the set of all MRS structures generated by the MRS semantic composition process (Manshadi et al., 2008) . CF-UR in its general form is not tractable. Therefore, we define a notion of coherence called heart-connectedness and show that all heart-connected CF-UR structures can be solved efficiently. We also show that heart-connected CF-UR covers the family of non-net structures, so CF-UR is in fact the first framework to address the non-net structures. In spite of that, CF-UR is quite restricted and does not allow for adding new constraints after semantic composition.", | |
| "cite_spans": [ | |
| { | |
| "start": 125, | |
| "end": 148, | |
| "text": "(Manshadi et al., 2009)", | |
| "ref_id": "BIBREF15" | |
| }, | |
| { | |
| "start": 306, | |
| "end": 329, | |
| "text": "(Manshadi et al., 2008)", | |
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| } | |
| ], | |
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| }, | |
| { | |
| "text": "In recent work, Koller et al. (2008) suggest using Regular Tree Grammars for scope underspecification, a probabilistic version of which could be used to find the best reading. The framework goes beyond the formalisms discussed in this paper and is expressively complete in Ebert (2005) 's sense of completeness, i.e. it is able to describe any subset of the readings of a UR. However, this power comes at the cost of exponential complexity. In practice, RTG is built on top of weak nets, benefiting from the compactness of this framework to remain tractable. Being a super set of weak net, super net provides a more powerful core for RTG. Koller and Thater (2010) address the problem of finding the weakest readings of a UR, which are those entailed by some reading(s), but not entailing any other reading of the UR. By only considering the weakest readings, the space of solutions will be dramatically reduced. Note that entailment using the weakest readings is sound but not complete.", | |
| "cite_spans": [ | |
| { | |
| "start": 16, | |
| "end": 36, | |
| "text": "Koller et al. (2008)", | |
| "ref_id": "BIBREF13" | |
| }, | |
| { | |
| "start": 273, | |
| "end": 285, | |
| "text": "Ebert (2005)", | |
| "ref_id": "BIBREF6" | |
| }, | |
| { | |
| "start": 639, | |
| "end": 663, | |
| "text": "Koller and Thater (2010)", | |
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| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "Related work", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "Weakly normal dominance graph brings many current constraint-based formalisms under a uniform framework, but its configurability is intractable in its general form. In this paper, we present a tractable subset of this framework. We prove that this subset, called super net, is a strict superset of weak net, a previously known tractable subset of the framework, and that it covers a family of coherent natural language sentences whose underspecified representation are known not to belong to weak nets.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Summary and Future work", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "As mentioned in Section 5, another extension of weak nets, downward connected nets, has been proposed by Koller and Thater (2007) , which addresses some of the limitations of weak nets, yet is unable to solve the known family of non-net structures. A thorough comparison between super nets and downward connected nets remains for future work.", | |
| "cite_spans": [ | |
| { | |
| "start": 105, | |
| "end": 129, | |
| "text": "Koller and Thater (2007)", | |
| "ref_id": "BIBREF10" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Summary and Future work", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "Another interesting property of super nets to be explored is how they compare to heart-connected graphs. Heart-connectedness has been introduced as a mathematical criterion for verifying the coherence of an underspecified representation within the framework of underspecification graph (Manshadi et al., 2009 ). Our early investigation shows that super nets may contain all heart-connected UGs. If this conjecture is true, super net would be broad enough to cover every coherent natural language sentence (under this notion of coherence). We leave a detailed investigation of this conjecture for the future.", | |
| "cite_spans": [ | |
| { | |
| "start": 286, | |
| "end": 308, | |
| "text": "(Manshadi et al., 2009", | |
| "ref_id": "BIBREF15" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Summary and Future work", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "The main difference is in the concept of solution in the two frameworks. See Section 4.3 for details.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We assume that there is no constraint edge between two nodes of the same fragment.", | |
| "cite_spans": [], | |
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| "section": "", | |
| "sec_num": null | |
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| { | |
| "text": "u dominates v in the directed graph G, iff u reaches v in G by a directed path.4 Here, we are referring to the nodes in TU,P by calling the nodes u and v in U . This is a sound strategy, as every node in U is mapped into a unique node in TU,P . The inverse is not true though, as every node (except the root) in TU,P corresponds to one hole and one label in U . Addressing TU,P 's nodes in this way is convenient, so we practice that throughout the paper.5 Given a directed graph G and the nodes u and v in G, u is said to be weakly connected to v (and vice versa), iff u and v are connected in the underlying undirected graph of G. A weakly connected graph is a graph in which every two nodes are weakly connected. Since weak connectedness is an equivalence relation, it partitions a directed graph into equivalent classes each of which is called a weakly connected component or WCC.6 Since fragments are ordered trees, solutions are ordered trees as well.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
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| "text": "Necessary conditions of freeness in a UG are not exactly the same as the ones in a weakly normal dominance graph, as depicted inBodirsky et al. (2004), because the definition of solution is different for the two frameworks (c.f. Section 4.3).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "sec_num": null | |
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| { | |
| "text": "Throughout this paper, by path we always mean a simple path, that is no node may be visited more than once on a path.9 Note that even though p1 is not a hypernormal path, there is another hypernormal path connecting the same two nodes 10 v is a dominance child of u in a UG U , if (u, v) is a constraint edge in U .", | |
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| }, | |
| { | |
| "text": "As discussed in Section 5, by defining the notion of downward connectedness,Koller and Thater (2007) address this issue of weak nets, at the expense of cubic time complexity.12 For simplicity, throughout this paper we have used the term non-net to refer to non-(weak net) UGs.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": "This work was support in part by NSF grant 1012205, and ONR grant N000141110417.", | |
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| "eq_spans": [], | |
| "section": "Acknowledgments", | |
| "sec_num": null | |
| } | |
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| "type_str": "figure", | |
| "uris": null, | |
| "text": "Two of the solutions to the UG inFigure 3.", | |
| "num": null | |
| }, | |
| "FIGREF4": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "(a) is a solution of this UG. Similarly, Figures 4(a,b) represent two solutions of the UG in Figure 3. The solutions in Figures 2 and 4 are all tree structures. This is because UGs inFigures 1 and 3are weakly connected. 5 Lemma 2 proves that this holds in general, that is: Proposition 1. Every solution of a weakly connected UG is a tree.", | |
| "num": null | |
| }, | |
| "FIGREF5": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "Recursively solving UGs.", | |
| "num": null | |
| }, | |
| "FIGREF6": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "UG for Illustration of hypernormal path.", | |
| "num": null | |
| }, | |
| "FIGREF7": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "Openness).(Thater, 2007) A node u of a fragment F is called an open node iff it has no outgoing constraint edge.For example, l inFigure 5(a) is an open label node. In Figure 2(b), h 2 is an open hole. We are finally ready to define super net. Definition 8 (Super net). A UG U is called a super net if for every fragment F rooted at l: D8a. F has at most one open node. D8b. If l 1 and l 2 are two dominance children of a hole h of F, then l 1 and l 2 are hypernormally connected in U \u2212h. D8c. \u2022 Case 1: F has no open hole.Every dominance child 10 of l is hypernormally connected to some hole of F in U\u2212l. \u2022 Case 2: F has an open hole.All dominance children of l, not hypernormally connected to a hole of F in U\u2212l, are hypernormally connected together.", | |
| "num": null | |
| }, | |
| "FIGREF8": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "Illustration of super net conditions. Definition 9 (Types of fragment). Following Definition 8, super net allows for three possible types of fragment: D9a. Open-root: Only the root is open (Figure 5a) D9b. Open-hole: Only a hole is open (Figure 2b) D9c. Closed: F There is no open node.(Figure 2a)", | |
| "num": null | |
| }, | |
| "FIGREF9": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "Proof of Proposition 5", | |
| "num": null | |
| }, | |
| "FIGREF10": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "5. Let h i be the hole of F connected to U i in U \u2212l, for i = 1 \u2022 \u2022 \u2022 m. (If for some k, U k is not connected to any hole of F in U \u2212l, let h k be the open hole of F .) 6. Build T by plugging the root of T i into h i , for i = 1 \u2022 \u2022 \u2022 m. 7. Return T .", | |
| "num": null | |
| }, | |
| "FIGREF11": { | |
| "type_str": "figure", | |
| "uris": null, | |
| "text": "(i). F has exactly one open node, so it satisfies condition (D8a).(ii). For every two holes of F , condition(D11b)guarantees that condition (D8b) holds. (iii). \u2022 Case 1) F has no open hole: Based on condition (D11a) the root of F is open, hence it has no dominance children. (D8c) trivially holds in this case. \u2022 Case 2) F has an open hole:", | |
| "num": null | |
| }, | |
| "TABREF0": { | |
| "html": null, | |
| "num": null, | |
| "content": "<table/>", | |
| "text": "ENUM and SAT algorithms (|U | = def |V | + |E|, where |V | = def |L| + |H|, and |E| = def |S| + |C|). Therefore SAT (and each branch of ENUM) run(s) in O(|L|.|U", | |
| "type_str": "table" | |
| } | |
| } | |
| } | |
| } |