{ "paper_id": "P03-1047", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:14:15.022149Z" }, "title": "Bridging the Gap Between Underspecification Formalisms: Minimal Recursion Semantics as Dominance Constraints", "authors": [ { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "", "affiliation": { "laboratory": "Programming Systems Lab Universit\u00e4t des Saarlandes", "institution": "", "location": {} }, "email": "niehren@ps.uni-sb.de" }, { "first": "Stefan", "middle": [], "last": "Thater", "suffix": "", "affiliation": {}, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "Minimal Recursion Semantics (MRS) is the standard formalism used in large-scale HPSG grammars to model underspecified semantics. We present the first provably efficient algorithm to enumerate the readings of MRS structures, by translating them into normal dominance constraints.", "pdf_parse": { "paper_id": "P03-1047", "_pdf_hash": "", "abstract": [ { "text": "Minimal Recursion Semantics (MRS) is the standard formalism used in large-scale HPSG grammars to model underspecified semantics. We present the first provably efficient algorithm to enumerate the readings of MRS structures, by translating them into normal dominance constraints.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "In the past few years there has been considerable activity in the development of formalisms for underspecified semantics (Alshawi and Crouch, 1992; Reyle, 1993; Bos, 1996; Copestake et al., 1999; Egg et al., 2001) . The common idea is to delay the enumeration of all readings for as long as possible. Instead, they work with a compact underspecified representation; readings are enumerated from this representation by need.", "cite_spans": [ { "start": 121, "end": 147, "text": "(Alshawi and Crouch, 1992;", "ref_id": "BIBREF0" }, { "start": 148, "end": 160, "text": "Reyle, 1993;", "ref_id": "BIBREF11" }, { "start": 161, "end": 171, "text": "Bos, 1996;", "ref_id": "BIBREF3" }, { "start": 172, "end": 195, "text": "Copestake et al., 1999;", "ref_id": "BIBREF5" }, { "start": 196, "end": 213, "text": "Egg et al., 2001)", "ref_id": "BIBREF7" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Minimal Recursion Semantics (MRS) (Copestake et al., 1999) is the standard formalism for semantic underspecification used in large-scale HPSG grammars (Pollard and Sag, 1994; . Despite this clear relevance, the most obvious questions about MRS are still open:", "cite_spans": [ { "start": 34, "end": 58, "text": "(Copestake et al., 1999)", "ref_id": "BIBREF5" }, { "start": 151, "end": 174, "text": "(Pollard and Sag, 1994;", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "1. Is it possible to enumerate the readings of MRS structures efficiently? No algorithm has been published so far. Existing implementations seem to be practical, even though the problem whether an MRS has a reading is NPcomplete (Althaus et al., 2003, Theorem 10 .1). 2. What is the precise relationship to other underspecification formalism? Are all of them the same, or else, what are the differences?", "cite_spans": [ { "start": 229, "end": 262, "text": "(Althaus et al., 2003, Theorem 10", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "We distinguish the sublanguages of MRS nets and normal dominance nets, and show that they can be intertranslated. This translation answers the first question: existing constraint solvers for normal dominance constraints can be used to enumerate the readings of MRS nets in low polynomial time.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "The translation also answers the second question restricted to pure scope underspecification. It shows the equivalence of a large fragment of MRSs and a corresponding fragment of normal dominance constraints, which in turn is equivalent to a large fragment of Hole Semantics (Bos, 1996) as proven in . Additional underspecified treatments of ellipsis or reinterpretation, however, are available for extensions of dominance constraint only (CLLS, the constraint language for lambda structures (Egg et al., 2001) ).", "cite_spans": [ { "start": 275, "end": 286, "text": "(Bos, 1996)", "ref_id": "BIBREF3" }, { "start": 492, "end": 510, "text": "(Egg et al., 2001)", "ref_id": "BIBREF7" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Our results are subject to a new proof technique which reduces reasoning about MRS structures to reasoning about weakly normal dominance constraints (Bodirsky et al., 2003) . The previous proof techniques for normal dominance constraints do not apply.", "cite_spans": [ { "start": 149, "end": 172, "text": "(Bodirsky et al., 2003)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "We define a simplified version of Minimal Recursion Semantics and discuss differences to the original definitions presented in (Copestake et al., 1999) .", "cite_spans": [ { "start": 127, "end": 151, "text": "(Copestake et al., 1999)", "ref_id": "BIBREF5" } ], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "MRS is a description language for formulas of first order object languages with generalized quantifiers. Underspecified representations in MRS consist of elementary predications and handle constraints. Roughly, elementary predications are object language formulas with \"holes\" into which other formulas can be plugged; handle constraints restrict the way these formulas can be plugged into each other. More formally, MRSs are formulas over the following vocabulary: 4. The symbol \u2264 for the outscopes relation.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "Formulas of MRS have three kinds of literals, the first two are called elementary predications (EPs) and the third handle constraints:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "1. h : P(x 1 , . . . , x n , h 1 , . . . , h m ) where n, m \u2265 0 2. h : Q x (h 1 , h 2 ) 3. h 1 \u2264 h 2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "Label positions are to the left of colons ':' and argument positions to the right. Let M be a set of literals. The label set lab(M) contains those handles of M that occur in label but not in argument position. The argument handle set arg(M) contains the handles of M that occur in argument but not in label position.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "M1 Every handle occurs at most once in label and at most once in argument position in M. M2 Handle constraints h 1 \u2264 h 2 in M always relate argument handles h 1 to labels h 2 of M. M3 For every constant (individual variable) x in argument position in M there is a unique literal of the form h :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Q x (h 1 , h 2 ) in M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "We call an MRS compact if it additionally satisfies:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "M4 Every handle of M occurs exactly once in an elementary predication of M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "We say that a handle h immediately outscopes a handle h in an MRS M iff there is an EP E in M such that h occurs in label and h in argument position of E. The outscopes relation is the reflexive, transitive closure of the immediate outscopes relation.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "every x student x read x,y some y book y {h 1 : every x (h 2 , h 4 ), h 3 : student(x), h 5 : some y (h 6 , h 8 ), h 7 : book(y), h 9 : read(x, y), h 2 \u2264 h 3 , h 6 \u2264 h 7 }", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Figure 1: MRS for \"Every student reads a book\".", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "An example MRS for the scopally ambiguous sentence \"Every student reads a book\" is given in Fig. 1 . We often represent MRSs by directed graphs whose nodes are the handles of the MRS. Elementary predications are represented by solid edges and handle constraints by dotted lines. Note that we make the relation between bound variables and their binders explicit by dotted lines (as from every x to read x,y ); redundant \"binding-edges\" that are subsumed by sequences of other edges are omitted however (from every x to student x for instance).", "cite_spans": [], "ref_spans": [ { "start": 92, "end": 98, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "A solution for an underspecified MRS is called a configuration, or scope-resolved MRS.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Definition 2 (Configuration). An MRS M is a configuration if it satisfies the following conditions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "C1 The graph of M is a tree of solid edges: handles don't properly outscope themselves or occur in different argument positions and all handles are pairwise connected by elementary predications. C2 If two EPs h : P(. . . , x, . . .) and h 0 :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Q x (h 1 , h 2 )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "belong to M, then h 0 outscopes h in M (so that the binding edge from h 0 to h is redundant). We call M a configuration for another MRS M if there exists some substitution \u03c3 : arg(M ) \u2192 lab(M ) which states how to identify argument handles of M with labels of M , so that:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "C3 M = {\u03c3(E) | E is EP in M }, and C4 \u03c3(h 1 ) outscopes h 2 in M, for all h 1 \u2264 h 2 \u2208 M .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "The value \u03c3(E) is obtained by substituting all argument handles in E, leaving all others unchanged.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "The MRS in Fig. 1 has precisely two configurations displayed in Fig. 2 which correspond to the two readings of the sentence. In this paper, we present an algorithm that enumerates the configurations of MRSs efficiently. Differences to Standard MRS. Our version departs from standard MRS in some respects. First, we assume that different EPs must be labeled with different handles, and that labels cannot be identified. In standard MRS, however, conjunctions are encoded by labeling different EPs with the same handle. These EP-conjunctions can be replaced in a preprocessing step introducing additional EPs that make conjunctions explicit.", "cite_spans": [], "ref_spans": [ { "start": 11, "end": 17, "text": "Fig. 1", "ref_id": null }, { "start": 64, "end": 70, "text": "Fig. 2", "ref_id": "FIGREF1" } ], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Second, our outscope constraints are slightly less restrictive than the original \"qeq-constraints.\" A handle h is qeq to a handle h in an MRS M, h = q h , if either h = h or a quantifier h :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Q x (h 1 , h 2 ) occurs in M and h 2 is qeq to h in M. Thus, h = q h im- plies h \u2264 h ,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "but not the other way round. We believe that the additional strength of qeq-constraints is not needed in practice for modeling scope. Recent work in semantic construction for HPSG (Copestake et al., 2001 ) supports our conjecture: the examples discussed there are compatible with our simplification.", "cite_spans": [ { "start": 180, "end": 203, "text": "(Copestake et al., 2001", "ref_id": "BIBREF6" } ], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Third, we depart in some minor details: we use sets instead of multi-sets and omit top-handles which are useful only during semantics construction.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS). An MRS is finite set M of MRS-literals such that:", "sec_num": null }, { "text": "Dominance constraints are a general framework for describing trees, and thus syntax trees of logical formulas. Dominance constraints are the core language underlying CLLS (Egg et al., 2001 ) which adds parallelism and binding constraints.", "cite_spans": [ { "start": 171, "end": 188, "text": "(Egg et al., 2001", "ref_id": "BIBREF7" } ], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "We assume a possibly infinite signature \u03a3 of function symbols with fixed arities and an infinite set Var of variables ranged over by X ,Y, Z. We write f , g for function symbols and ar( f ) for the arity of f .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntax and Semantics", "sec_num": "3.1" }, { "text": "A dominance constraint \u03d5 is a conjunction of dominance, inequality, and labeling literals of the following forms where ar( f ) = n:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntax and Semantics", "sec_num": "3.1" }, { "text": "\u03d5 ::= X * Y | X = Y | X : f (X 1 , . . . , X n ) | \u03d5 \u2227 \u03d5", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntax and Semantics", "sec_num": "3.1" }, { "text": "Dominance constraints are interpreted over finite constructor trees, i.e. ground terms constructed from the function symbols in \u03a3. We identify ground terms with trees that are rooted, ranked, edge-ordered and labeled. A solution for a dominance constraint consists of a tree \u03c4 and a variable assignment \u03b1 that maps variables to nodes of \u03c4 such that all constraints are satisfied: a labeling literal X : f (X 1 , . . . , X n ) is satisfied iff the node \u03b1(X ) is labeled with f and has daughters \u03b1(X 1 ), . . . , \u03b1(X n ) in this order; a dominance literal X * Y is satisfied iff \u03b1(X ) is an ancestor of \u03b1(Y ) in \u03c4; and an inequality literal X = Y is satisfied iff \u03b1(X ) and \u03b1(Y ) are distinct nodes.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntax and Semantics", "sec_num": "3.1" }, { "text": "Note that solutions may contain additional material. The tree f (a, b), for instance, satisfies the constraint Y : a \u2227 Z :b.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Syntax and Semantics", "sec_num": "3.1" }, { "text": "The satisfiability problem of arbitrary dominance constraints is NP-complete ) in general. However, Althaus et al. (2003) identify a natural fragment of so called normal dominance constraints, which have a polynomial time satisfiability problem. Bodirsky et al. (2003) generalize this notion to weakly normal dominance constraints.", "cite_spans": [ { "start": 100, "end": 121, "text": "Althaus et al. (2003)", "ref_id": "BIBREF1" }, { "start": 246, "end": 268, "text": "Bodirsky et al. (2003)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Normality and Weak Normality", "sec_num": "3.2" }, { "text": "We call a variable a hole of \u03d5 if it occurs in argument position in \u03d5 and a root of \u03d5 otherwise.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normality and Weak Normality", "sec_num": "3.2" }, { "text": "A dominance constraint \u03d5 is normal (and compact) if it satisfies the following conditions. N1 (a) each variable of \u03d5 occurs at most once in the labeling literals of \u03d5. (b) each variable of \u03d5 occurs at least once in the labeling literals of \u03d5. N2 for distinct roots X and Y of \u03d5, X = Y is in \u03d5. N3 (a) if X * Y occurs in \u03d5, Y is a root in \u03d5.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "(b) if X * Y occurs in \u03d5, X is a hole in \u03d5. A dominance constraint is weakly normal if it satisfies all above properties except for N1(b) and N3(b).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "The idea behind (weak) normality is that the constraint graph (see below) of a dominance constraint consists of solid fragments which are connected by dominance constraints; these fragments may not properly overlap in solutions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "Note that Definition 3 always imposes compactness, meaning that the heigth of solid fragments is at most one. As for MRS, this is not a serious restriction, since more general weakly normal dominance constraints can be compactified, provided that dominance links relate either roots or holes with roots. Dominance Graphs. We often represent dominance constraints as graphs. A dominance graph is the directed graph (V, * ). The graph of a weakly normal constraint \u03d5 is defined as follows: The nodes of the graph of \u03d5 are the variables of \u03d5. A labeling literal X : f (X 1 , . . . , X n ) of \u03d5 contributes tree edges (X , X i ) \u2208 for 1 \u2264 i \u2264 n that we draw as X X i ; we freely omit the label f and the edge order in the graph. A dominance literal X * Y contributes a dominance edge (X ,Y ) \u2208 * that we draw as X Y . Inequality literals in \u03d5 are also omitted in the graph. f a g For example, the constraint graph on the right represents the dominance constraint X :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "f (X ) \u2227Y : g(Y ) \u2227 X * Z \u2227 Y * Z \u2227 Z :a \u2227 X =Y \u2227 X =Z \u2227Y =Z.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "A dominance graph is weakly normal or a wndgraph if it does not contain any forbidden subgraphs:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "Dominance graphs of a weakly normal dominance constraints are clearly weakly normal.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "Solved Forms and Configurations. The main difference between MRS and dominance constraints lies in their notion of interpretation: solutions versus configurations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "Every satisfiable dominance constraint has infinitely many solutions. Algorithms for dominance constraints therefore do not enumerate solutions but solved forms. We say that a dominance constraint is in solved form iff its graph is in solved form. A wndgraph \u03a6 is in solved form iff \u03a6 is a forest. The solved forms of \u03a6 are solved forms \u03a6 that are more specific than \u03a6, i.e. \u03a6 and \u03a6 differ only in their dominance edges and the reachability relation of \u03a6 extends the reachability of \u03a6 . A minimal solved form of \u03a6 is a solved form of \u03a6 that is minimal with respect to specificity.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "The notion of configurations from MRS applies to dominance constraints as well. Here, a configuration is a dominance constraint whose graph is a tree without dominance edges. A configuration of a constraint \u03d5 is a configuration that solves \u03d5 in the obvious sense. Simple solved forms are tree-shaped solved forms where every hole has exactly one outgoing dominance edge. Lemma 1. Simple solved forms and configurations correspond: Every simple solved form has exactly one configuration, and for every configuration there is exactly one solved form that it configures. Unfortunately, Lemma 1 does not extend to minimal as opposed to simple solved forms: there are minimal solved forms without configurations. The constraint on the right of Fig. 3 , for instance, has no configuration: the hole of L1 would have to be filled twice while the right hole of L2 cannot be filled.", "cite_spans": [], "ref_spans": [ { "start": 739, "end": 745, "text": "Fig. 3", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Definition 3.", "sec_num": null }, { "text": "We next map (compact) MRSs to weakly normal dominance constraints so that configurations are preserved. Note that this translation is based on a non-standard semantics for dominance constraints, namely configurations. We address this problem in the following sections.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Representing MRSs", "sec_num": "4" }, { "text": "The translation of an MRS M to a dominance constraint \u03d5 M is quite trivial. The variables of \u03d5 M are the handles of M and its literal set is: This weak correctness property follows straightforwardly from the analogy in the definitions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Representing MRSs", "sec_num": "4" }, { "text": "{h : P x 1 ,...,x n (h 1 , . . .) | h : P(x 1 , . . . , x n , h 1 , . . .) \u2208 M} \u222a{h : Q x (h 1 , h 2 ) | h : Q x (h 1 , h 2 ) \u2208 M} \u222a{h 1 * h 2 | h 1 \u2264 h 2 \u2208 M} \u222a{h * h 0 | h : Q x (h 1 , h 2 ), h 0 : P(. . . , x, . . .) \u2208 M} \u222a{h =h | h,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Representing MRSs", "sec_num": "4" }, { "text": "We recall an algorithm from (Bodirsky et al., 2003) that efficiently enumerates all minimal solved forms of wnd-graphs or constraints. All results of this section are proved there.", "cite_spans": [ { "start": 28, "end": 51, "text": "(Bodirsky et al., 2003)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Constraint Solving", "sec_num": "5" }, { "text": "The algorithm can be used to enumerate configurations for a large subclass of MRSs, as we will see in Section 6. But equally importantly, this algorithm provides a powerful proof method for reasoning about solved forms and configurations on which all our results rely.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Constraint Solving", "sec_num": "5" }, { "text": "Two nodes X and Y of a wnd-graph \u03a6 = (V, E) are weakly connected if there is an undirected path from X to Y in (V, E). We call \u03a6 weakly connected if all its nodes are weakly connected. A weakly connected component (wcc) of \u03a6 is a maximal weakly connected subgraph of \u03a6. The wccs of \u03a6 = (V, E) form proper partitions of V and E. Proposition 2. The graph of a solved form of a weakly connected wnd-graph is a tree.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Weak Connectedness", "sec_num": "5.1" }, { "text": "The enumeration algorithm is based on the notion of freeness.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Freeness", "sec_num": "5.2" }, { "text": "Definition 4. A node X of a wnd-graph \u03a6 is called free in \u03a6 if there exists a solved form of \u03a6 whose graph is a tree with root X .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Freeness", "sec_num": "5.2" }, { "text": "A weakly connected wnd-graph without free nodes is unsolvable. Otherwise, it has a solved form whose graph is a tree (Prop. 2) and the root of this tree is free in \u03a6.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Freeness", "sec_num": "5.2" }, { "text": "Given a set of nodes V \u2286 V , we write \u03a6| V for the restriction of \u03a6 to nodes in V and edges in V \u00d7V . The following lemma characterizes freeness: Lemma 2. A wnd-graph \u03a6 with free node X satisfies the freeness conditions:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Freeness", "sec_num": "5.2" }, { "text": "F1 node X has indegree zero in graph \u03a6, and F2 no distinct children Y and Y of X in \u03a6 that are linked to X by immediate dominance edges are weakly connected in the remainder \u03a6| V \\{X} .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Freeness", "sec_num": "5.2" }, { "text": "The algorithm for enumerating the minimal solved forms of a wnd-graph (or equivalently constraint) is given in Fig. 4 . We illustrate the algorithm for the problematic wnd-graph \u03a6 in Fig. 3 . The graph of \u03a6 is weakly connected, so that we can call solve(\u03a6). This procedure guesses topmost fragments in solved forms of \u03a6 (which always exist by Prop. 2). The only candidates are L1 or L2 since L3 and L4 have incoming dominance edges, which violates F1. Let us choose the fragment L2 to be topmost. The graph which remains when removing L2 is still weakly connected. It has a single minimal solved form computed by a recursive call of the solver, where L1 dominates L3 and L4. The solved form of the restricted graph is then put below the left hole of L2, since it is connected to this hole. As a result, we obtain the solved form on the right of Fig. 3 .", "cite_spans": [], "ref_spans": [ { "start": 111, "end": 117, "text": "Fig. 4", "ref_id": "FIGREF3" }, { "start": 183, "end": 189, "text": "Fig. 3", "ref_id": "FIGREF2" }, { "start": 845, "end": 851, "text": "Fig. 3", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Algorithm", "sec_num": "5.3" }, { "text": "Theorem 1. The function solved-form(\u03a6) computes all minimal solved forms of a weakly normal dominance graph \u03a6; it runs in quadratic time per solved form.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Algorithm", "sec_num": "5.3" }, { "text": "Next, we explain how to encode a large class of MRSs into wnd-constraints such that configurations correspond precisely to minimal solved forms. The result of the translation will indeed be normal.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Full Translation", "sec_num": "6" }, { "text": "The naive representation of MRSs as weakly normal dominance constraints is only correct in a weak sense. The encoding fails in that some MRSs which have no configurations are mapped to solvable wndconstraints. For instance, this holds for the MRS on the right in Fig 3. We cannot even hope to translate arbitrary MRSs correctly into wnd-constraints: the configurability problem of MRSs is NP-complete, while satisfiability of wnd-constraints can be solved in polynomial time. Instead, we introduce the sublanguages of MRS-nets and equivalent wnd-nets, and show that they can be intertranslated in quadratic time.", "cite_spans": [], "ref_spans": [ { "start": 263, "end": 269, "text": "Fig 3.", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Problems and Examples", "sec_num": "6.1" }, { "text": "Let \u03a6 1 , . . . , \u03a6 k be the wccs of", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "\u03a6 = (V, E) Let (V i , E i ) be the result of solve(\u03a6 i ) return (V, \u222a k i=1 E i ) solve(\u03a6) \u2261 precond: \u03a6 = (V, * )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "is weakly connected choose a node X satisfying (F1) and (F2) in \u03a6 else fail Let Y 1 , . . . ,Y n be all nodes s.t. X Y i Let \u03a6 1 , . . . , \u03a6 k be the weakly connected components of", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "\u03a6| V \u2212{X,Y 1 ,...,Y n } Let (W j , E j )", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "be the result of solve(\u03a6 j ), and ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "X j \u2208 W j its root return (V, \u222a k j=1 E j \u222a \u222a * 1 \u222a * 2 ) where * 1 = {(Y i , X j ) | \u2203X : (Y i , X ) \u2208 * \u2227 X \u2208 W j }, * 2 = {(X , X j ) | \u00ac\u2203X : (Y i , X ) \u2208 * \u2227 X \u2208 W j }", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "solved-form(\u03a6) \u2261", "sec_num": null }, { "text": "A hypernormal path (Althaus et al., 2003) in a wndgraph is a sequence of adjacent edges that does not traverse two outgoing dominance edges of some hole X in sequence, i.e. a wnd-graph without situations Y 1 X Y 2 . A dominance net \u03a6 is a weakly normal dominance constraint whose fragments all satisfy one of the three schemas in Fig. 5 . MRS-nets can be defined analogously. This means that all roots of \u03a6 are labeled in \u03a6, and that all fragments X : f (X 1 , . . . , X n ) of \u03a6 satisfy one of the following three conditions: strong. n \u2265 0 and for all Y \u2208 {X 1 , . . . , X n } there exists a unique Z such that Y * Z in \u03a6, and there exists no Z such that X * Z in \u03a6. weak. n \u2265 1 and for all Y \u2208 {X 1 , . . . , X n\u22121 , X } there exists a unique Z such that Y * Z in \u03a6, and there exists no Z such that X n * Z in \u03a6.", "cite_spans": [ { "start": 19, "end": 41, "text": "(Althaus et al., 2003)", "ref_id": "BIBREF1" } ], "ref_spans": [ { "start": 330, "end": 336, "text": "Fig. 5", "ref_id": null } ], "eq_spans": [], "section": "Dominance and MRS-Nets", "sec_num": "6.2" }, { "text": "island. n = 1 and all variables in {Y | X 1 * Y } are connected by a hypernormal path in the graph of the restricted constraint \u03a6 |V \u2212{X 1 } , and there exists no Z such that X * Z in \u03a6.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance and MRS-Nets", "sec_num": "6.2" }, { "text": "The requirement of hypernormal connections in islands replaces the notion of chain-connectedness in , which fails to apply to dominance constraints with weak dominance edges.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance and MRS-Nets", "sec_num": "6.2" }, { "text": "For ease of presentation, we restrict ourselves to a simple version of island fragments. In general, we should allow for island fragments with n > 1.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance and MRS-Nets", "sec_num": "6.2" }, { "text": "Dominance nets are wnd-constraints. We next translate dominance nets \u03a6 to normal dominance constraints \u03a6 so that \u03a6 has a configuration iff \u03a6 is satisfiable. The trick is to normalize weak dominance edges. The normalization norm(\u03a6) of a weakly normal dominance constraint \u03a6 is obtained by converting all root-to-root dominance literals X * Y as follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "X * Y \u21d2 X n * Y", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "if X roots a fragment of \u03a6 that satisfies schema weak of net fragments. If \u03a6 is a dominance net then norm(\u03a6) is indeed a normal dominance net.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "Theorem 2. The configurations of a weakly connected dominance net \u03a6 correspond bijectively to the minimal solved forms of its normalization norm(\u03a6).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "For illustration, consider the problematic wndconstraint \u03a6 on the left of Fig. 3 . \u03a6 has two minimal solved forms with top-most fragments L1 and L2 respectively. The former can be configured, in contrast to the later which is drawn on the right of Fig. 3 .", "cite_spans": [], "ref_spans": [ { "start": 74, "end": 80, "text": "Fig. 3", "ref_id": "FIGREF2" }, { "start": 248, "end": 254, "text": "Fig. 3", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "Normalizing \u03a6 has an interesting consequence: norm(\u03a6) has (in contrast to \u03a6) a single minimal solved form with L1 on top. Indeed, norm(\u03a6) cannot be satisfied while placing L2 topmost. Our algorithm detects this correctly: the normalization of fragment L2 is not free in norm(\u03a6) since it violates property", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Normalizing Dominance Nets", "sec_num": "6.3" }, { "text": "The proof of Theorem 2 captures the rest of this section. We show in a first step (Prop. 3) that the configurations are preserved when normalizing weakly connected and satisfiable nets. In the second step, we show that minimal solved forms of normalized nets, and thus of norm(\u03a6), can always be configured (Prop. 4). Corollary 1. Configurability of weakly connected MRS-nets can be decided in polynomial time; configurations of weakly connected MRS-nets can be enumerated in quadratic time per configuration.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "F2.", "sec_num": null }, { "text": "Most importantly, nets can be recursively decomposed into nets as long as they have configurations:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Lemma 3. If a dominance net \u03a6 has a configuration whose top-most fragment is X : f (X 1 , . . . , X n ), then the restriction \u03a6 |V \u2212{X,X 1 ,...,X n } is a dominance net.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Note that the restriction of the problematic net \u03a6 by L2 on the left in Fig. 3 is not a net. This does not contradict the lemma, as \u03a6 does not have a configuration with top-most fragment L2.", "cite_spans": [], "ref_spans": [ { "start": 72, "end": 78, "text": "Fig. 3", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Proof. First note that as X is free in \u03a6 it cannot have incoming edges (condition F1). This means that the restriction deletes only dominance edges that depart from nodes in {X , X 1 , . . . , X n }. Other fragments thus only lose ingoing dominance edges by normality condition N3. Such deletions preserve the validity of the schemas weak and strong.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "The island schema is more problematic. We have to show that the hypernormal connections in this schema can never be cut. So suppose that Y : f (Y 1 ) is an island fragment with outgoing dominance edges Y 1 * Z 1 and Y 1 * Z 2 , so that Z 1 and Z 2 are connected by some hypernormal path traversing the deleted fragment X : f (X 1 , . . . , X n ). We distinguish the three possible schemata for this fragment: Figure 6 : Traversals through fragments of free roots strong: since X does not have incoming dominance edges, there is only a single non-trival kind of traversal, drawn in Fig. 6(a) . But such traversals contradict the freeness of X according to F2. weak: there is one other way of traversing weak fragments, shown in Fig. 6(b) . Let X * Y be the weak dominance edge. The traversal proves that Y belongs to the weakly connected components of one of the X i , so the \u03a6 \u2227 X n * Y is unsatisfiable. This shows that the hole X n cannot be identified with any root, i.e. \u03a6 does not have any configuration in contrast to our assumption. island: free island fragments permit one single nontrivial form of traversals, depicted in Fig. 6(c) . But such traversals are not hypernormal. Proof. Let C be a configuration of \u03a6. We show that it also configures norm(\u03a6). Let S be the simple solved form of \u03a6 that is configured by C (Lemma 1), and S be a minimal solved form of \u03a6 which is more general than S.", "cite_spans": [], "ref_spans": [ { "start": 409, "end": 417, "text": "Figure 6", "ref_id": null }, { "start": 581, "end": 590, "text": "Fig. 6(a)", "ref_id": null }, { "start": 727, "end": 736, "text": "Fig. 6(b)", "ref_id": null }, { "start": 1131, "end": 1140, "text": "Fig. 6(c)", "ref_id": null } ], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Let X : f (Y 1 , . . . ,Y n ) be the top-most fragment of the tree S. This fragment must also be the top-most fragment of S , which is a tree since \u03a6 is assumed to be weakly connected (Prop. 2). S is constructed by our algorithm (Theorem 1), so that the evaluation of solve(\u03a6) must choose X as free root in \u03a6.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Since \u03a6 is a net, some literal X : f (Y 1 , . . . ,Y n ) must belong to \u03a6. Let \u03a6 = \u03a6 |{X,Y 1 ,...,Y n } be the restriction of \u03a6 to the lower fragments. The weakly connected components of all Y 1 , . . ., Y n\u22121 must be pairwise disjoint by F2 (which holds by Lemma 2 since X is free in \u03a6). The X -fragment of net \u03a6 must satisfy one of three possible schemata of net fragments: weak fragments: there exists a unique weak dominance edge X * Z in \u03a6 and a unique hole Y n without outgoing dominance edges. The variable Z must be a root in \u03a6 and thus be labeled. If Z is equal to X then \u03a6 is unsatisfiable by normality condition N2, which is impossible. Hence, Z occurs in the restriction \u03a6 but not in the weakly connected components of any Y 1 , . . ., Y n\u22121 . Otherwise, the minimal solved form S could not be configured since the hole Y n could not be identified with any root. Furthermore, the root of the Z-component must be identified with Y n in any configuration of \u03a6 with root X . Hence, C satisfies Y n * Z which is add by normalization.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "The restriction \u03a6 must be a dominance net by Lemma 3, and hence, all its weakly connected components are nets. For all 1 \u2264 i \u2264 n \u2212 1, the component of Y i in \u03a6 is configured by the subtree of C at node Y i , while the subtree of C at node Y n configures the component of Z in \u03a6 . The induction hypothesis yields that the normalizations of all these components are configured by the respective subconfigurations of C. Hence, norm(\u03a6) is configured by C. strong or island fragments are not altered by normalization, so we can recurse to the lower fragments (if there exist any).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Proposition 4. Minimal solved forms of normal, weakly connected dominance nets have configurations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "Proof. By induction over the construction of minimal solved forms, we can show that all holes of minimal solved forms have a unique outgoing dominance edge at each hole. Furthermore, all minimal solved forms are trees since we assumed connectedness (Prop.2). Thus, all minimal solved forms are simple, so they have configurations (Lemma 1).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Correctness Proof", "sec_num": "6.4" }, { "text": "We have related two underspecification formalism, MRS and normal dominance constraints. We have distinguished the sublanguages of MRS-nets and normal dominance nets that are sufficient to model scope underspecification, and proved their equivalence. Thereby, we have obtained the first provably efficient algorithm to enumerate the readings of underspecified semantic representations in MRS. Our encoding has the advantage that researchers interested in dominance constraints can benefit from the large grammar resources of MRS. This requires further work in order to deal with unrestricted versions of MRS used in practice. Conversely, one can now lift the additional modeling power of CLLS to MRS.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" } ], "back_matter": [], "bib_entries": { "BIBREF0": { "ref_id": "b0", "title": "Monotonic semantic interpretation", "authors": [ { "first": "H", "middle": [], "last": "Alshawi", "suffix": "" }, { "first": "R", "middle": [], "last": "Crouch", "suffix": "" } ], "year": 1992, "venue": "Proc. 30th ACL", "volume": "", "issue": "", "pages": "32--39", "other_ids": {}, "num": null, "urls": [], "raw_text": "H. Alshawi and R. Crouch. 1992. Monotonic semantic interpretation. In Proc. 30th ACL, pages 32-39.", "links": null }, "BIBREF1": { "ref_id": "b1", "title": "An efficient graph algorithm for dominance constraints", "authors": [ { "first": "E", "middle": [], "last": "Althaus", "suffix": "" }, { "first": "D", "middle": [], "last": "Duchier", "suffix": "" }, { "first": "A", "middle": [], "last": "Koller", "suffix": "" }, { "first": "K", "middle": [], "last": "Mehlhorn", "suffix": "" }, { "first": "J", "middle": [], "last": "Niehren", "suffix": "" }, { "first": "S", "middle": [], "last": "Thiel", "suffix": "" } ], "year": 2003, "venue": "Journal of Algorithms", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "E. Althaus, D. Duchier, A. Koller, K. Mehlhorn, J. Niehren, and S. Thiel. 2003. An efficient graph algorithm for dominance constraints. Journal of Algo- rithms. In press.", "links": null }, "BIBREF2": { "ref_id": "b2", "title": "An efficient algorithm for weakly normal dominance constraints. Available at www.ps.uni-sb", "authors": [ { "first": "Manuel", "middle": [], "last": "Bodirsky", "suffix": "" }, { "first": "Denys", "middle": [], "last": "Duchier", "suffix": "" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "" }, { "first": "Sebastian", "middle": [], "last": "Miele", "suffix": "" } ], "year": 2003, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Manuel Bodirsky, Denys Duchier, Joachim Niehren, and Sebastian Miele. 2003. An efficient algorithm for weakly normal dominance constraints. Available at www.ps.uni-sb.de/Papers.", "links": null }, "BIBREF3": { "ref_id": "b3", "title": "Predicate logic unplugged", "authors": [ { "first": "Johan", "middle": [], "last": "Bos", "suffix": "" } ], "year": 1996, "venue": "Amsterdam Colloquium", "volume": "", "issue": "", "pages": "133--143", "other_ids": {}, "num": null, "urls": [], "raw_text": "Johan Bos. 1996. Predicate logic unplugged. In Amster- dam Colloquium, pages 133-143.", "links": null }, "BIBREF4": { "ref_id": "b4", "title": "An opensource grammar development environment and broadcoverage English grammar using HPSG", "authors": [ { "first": "Ann", "middle": [], "last": "Copestake", "suffix": "" }, { "first": "Dan", "middle": [], "last": "Flickinger", "suffix": "" } ], "year": null, "venue": "Conference on Language Resources and Evaluation", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Ann Copestake and Dan Flickinger. An open- source grammar development environment and broad- coverage English grammar using HPSG. In Confer- ence on Language Resources and Evaluation.", "links": null }, "BIBREF5": { "ref_id": "b5", "title": "Minimal Recursion Semantics: An Introduction", "authors": [ { "first": "Ann", "middle": [], "last": "Copestake", "suffix": "" }, { "first": "Dan", "middle": [], "last": "Flickinger", "suffix": "" } ], "year": 1999, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Ann Copestake, Dan Flickinger, Ivan Sag, and Carl Pol- lard. 1999. Minimal Recursion Semantics: An Intro- duction. Manuscript, Stanford University.", "links": null }, "BIBREF6": { "ref_id": "b6", "title": "An algebra for semantic construction in constraint-based grammars", "authors": [ { "first": "Ann", "middle": [], "last": "Copestake", "suffix": "" }, { "first": "Alex", "middle": [], "last": "Lascarides", "suffix": "" }, { "first": "Dan", "middle": [], "last": "Flickinger", "suffix": "" } ], "year": 2001, "venue": "Proceedings of the 39th ACL", "volume": "", "issue": "", "pages": "132--139", "other_ids": {}, "num": null, "urls": [], "raw_text": "Ann Copestake, Alex Lascarides, and Dan Flickinger. 2001. An algebra for semantic construction in constraint-based grammars. In Proceedings of the 39th ACL, pages 132-139, Toulouse, France.", "links": null }, "BIBREF7": { "ref_id": "b7", "title": "The Constraint Language for Lambda Structures. Logic, Language, and Information", "authors": [ { "first": "Markus", "middle": [], "last": "Egg", "suffix": "" }, { "first": "Alexander", "middle": [], "last": "Koller", "suffix": "" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "" } ], "year": 2001, "venue": "", "volume": "10", "issue": "", "pages": "457--485", "other_ids": {}, "num": null, "urls": [], "raw_text": "Markus Egg, Alexander Koller, and Joachim Niehren. 2001. The Constraint Language for Lambda Struc- tures. Logic, Language, and Information, 10:457-485.", "links": null }, "BIBREF8": { "ref_id": "b8", "title": "Dominance constraints: Algorithms and complexity", "authors": [ { "first": "Alexander", "middle": [], "last": "Koller", "suffix": "" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "" }, { "first": "Ralf", "middle": [], "last": "Treinen", "suffix": "" } ], "year": 2001, "venue": "LACL'98", "volume": "2014", "issue": "", "pages": "106--125", "other_ids": {}, "num": null, "urls": [], "raw_text": "Alexander Koller, Joachim Niehren, and Ralf Treinen. 2001. Dominance constraints: Algorithms and com- plexity. In LACL'98, volume 2014 of LNAI, pages 106-125.", "links": null }, "BIBREF9": { "ref_id": "b9", "title": "Bridging the gap between underspecification formalisms: Hole semantics as dominance constraints", "authors": [ { "first": "Alexander", "middle": [], "last": "Koller", "suffix": "" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "" }, { "first": "Stefan", "middle": [], "last": "Thater", "suffix": "" } ], "year": 2003, "venue": "EACL'03", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Alexander Koller, Joachim Niehren, and Stefan Thater. 2003. Bridging the gap between underspecification formalisms: Hole semantics as dominance constraints. In EACL'03, April. In press.", "links": null }, "BIBREF10": { "ref_id": "b10", "title": "Head-driven Phrase Structure Grammar", "authors": [ { "first": "Carl", "middle": [], "last": "Pollard", "suffix": "" }, { "first": "Ivan", "middle": [], "last": "Sag", "suffix": "" } ], "year": 1994, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Carl Pollard and Ivan Sag. 1994. Head-driven Phrase Structure Grammar. University of Chicago Press.", "links": null }, "BIBREF11": { "ref_id": "b11", "title": "Dealing with ambiguities by underspecification: Construction, representation and deduction", "authors": [ { "first": "Uwe", "middle": [], "last": "Reyle", "suffix": "" } ], "year": 1993, "venue": "Journal of Semantics", "volume": "", "issue": "1", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Uwe Reyle. 1993. Dealing with ambiguities by under- specification: Construction, representation and deduc- tion. Journal of Semantics, 10(1).", "links": null } }, "ref_entries": { "FIGREF0": { "uris": null, "type_str": "figure", "num": null, "text": "(a) A set of function symbols written as P. (b) A set of quantifier symbols ranged over by Q (such as every and some). Pairs Q x are further function symbols (the variable binders of x in the object language)." }, "FIGREF1": { "uris": null, "type_str": "figure", "num": null, "text": "every x student x some y book y read x,y some y book y every x student x read x,y Graphs of Configurations." }, "FIGREF2": { "uris": null, "type_str": "figure", "num": null, "text": "A dominance constraint (left) with a minimal solved form (right) that has no configuration." }, "FIGREF3": { "uris": null, "type_str": "figure", "num": null, "text": "Enumerating the minimal solved-forms of a wnd-graph." }, "FIGREF5": { "uris": null, "type_str": "figure", "num": null, "text": "A configuration of a weakly connected dominance net \u03a6 configures its normalization norm(\u03a6), and vice versa of course." }, "TABREF0": { "type_str": "table", "text": "Variables. An infinite set of variables ranged over by h. Variables are also called handles. 2. Constants. An infinite set of constants ranged over by x, y, z. Constants are the individual variables of the object language. 3. Function symbols.", "num": null, "content": "", "html": null }, "TABREF1": { "type_str": "table", "text": "h in distinct label positions of M} The translation of a compact MRS M into a weakly normal dominance constraint \u03d5 M preserves configurations.", "num": null, "content": "
Compact MRSs M are clearly translated into (com-
pact) weakly normal dominance constraints. Labels
of M become roots in \u03d5 M while argument handles
become holes. Weak root-to-root dominance literals
are needed to encode variable binding condition
", "html": null } } } }