{ "paper_id": "P04-1032", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T08:44:14.000153Z" }, "title": "Minimal Recursion Semantics as Dominance Constraints: Translation, Evaluation, and Analysis", "authors": [ { "first": "Ruth", "middle": [], "last": "Fuchss", "suffix": "", "affiliation": { "laboratory": "", "institution": "Saarland University", "location": { "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "fuchss@coli.uni-sb.de" }, { "first": "Alexander", "middle": [], "last": "Koller", "suffix": "", "affiliation": { "laboratory": "", "institution": "Saarland University", "location": { "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "koller@coli.uni-sb.de" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "", "affiliation": { "laboratory": "", "institution": "INRIA Futurs", "location": { "settlement": "Lille", "country": "France" } }, "email": "" }, { "first": "Stefan", "middle": [], "last": "Thater", "suffix": "", "affiliation": { "laboratory": "", "institution": "Saarland University", "location": { "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "We show that a practical translation of MRS descriptions into normal dominance constraints is feasible. We start from a recent theoretical translation and verify its assumptions on the outputs of the English Resource Grammar (ERG) on the Redwoods corpus. The main assumption of the translationthat all relevant underspecified descriptions are nets-is validated for a large majority of cases; all non-nets computed by the ERG seem to be systematically incomplete.", "pdf_parse": { "paper_id": "P04-1032", "_pdf_hash": "", "abstract": [ { "text": "We show that a practical translation of MRS descriptions into normal dominance constraints is feasible. We start from a recent theoretical translation and verify its assumptions on the outputs of the English Resource Grammar (ERG) on the Redwoods corpus. The main assumption of the translationthat all relevant underspecified descriptions are nets-is validated for a large majority of cases; all non-nets computed by the ERG seem to be systematically incomplete.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Underspecification is the standard approach to dealing with scope ambiguity (Alshawi and Crouch, 1992; Pinkal, 1996) . The readings of underspecified expressions are represented by compact and concise descriptions, instead of being enumerated explicitly. Underspecified descriptions are easier to derive in syntax-semantics interfaces (Egg et al., 2001; Copestake et al., 2001) , useful in applications such as machine translation (Copestake et al., 1995) , and can be resolved by need.", "cite_spans": [ { "start": 76, "end": 102, "text": "(Alshawi and Crouch, 1992;", "ref_id": "BIBREF0" }, { "start": 103, "end": 116, "text": "Pinkal, 1996)", "ref_id": "BIBREF13" }, { "start": 335, "end": 353, "text": "(Egg et al., 2001;", "ref_id": "BIBREF8" }, { "start": 354, "end": 377, "text": "Copestake et al., 2001)", "ref_id": "BIBREF5" }, { "start": 431, "end": 455, "text": "(Copestake et al., 1995)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Two important underspecification formalisms in the recent literature are Minimal Recursion Semantics (MRS) (Copestake et al., 2004) and dominance constraints (Egg et al., 2001) . MRS is the underspecification language which is used in large-scale HPSG grammars, such as the English Resource Grammar (ERG) (Copestake and Flickinger, 2000) . The main advantage of dominance constraints is that they can be solved very efficiently (Althaus et al., 2003; Bodirsky et al., 2004) . Niehren and Thater (2003) defined, in a theoretical paper, a translation from MRS into normal dominance constraints. This translation clarified the precise relationship between these two related formalisms, and made the powerful meta-theory of dominance constraints accessible to MRS. Their goal was to also make the large grammars for MRS and the efficient constraint solvers for dominance constraints available to the other formalism.", "cite_spans": [ { "start": 107, "end": 131, "text": "(Copestake et al., 2004)", "ref_id": "BIBREF6" }, { "start": 158, "end": 176, "text": "(Egg et al., 2001)", "ref_id": "BIBREF8" }, { "start": 305, "end": 337, "text": "(Copestake and Flickinger, 2000)", "ref_id": "BIBREF3" }, { "start": 428, "end": 450, "text": "(Althaus et al., 2003;", "ref_id": "BIBREF1" }, { "start": 451, "end": 473, "text": "Bodirsky et al., 2004)", "ref_id": "BIBREF2" }, { "start": 476, "end": 501, "text": "Niehren and Thater (2003)", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "However, Niehren and Thater made three technical assumptions:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "1. that EP-conjunction can be resolved in a preprocessing step; 2. that the qeq relation in MRS is simply dominance; 3. and (most importantly) that all linguistically correct and relevant MRS expressions belong to a certain class of constraints called nets.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "This means that it is not obvious whether their result can be immediately applied to the output of practical grammars like the ERG.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this paper, we evaluate the truth of these assumptions on the MRS expressions which the ERG computes for the sentences in the Redwoods Treebank (Oepen et al., 2002) . The main result of our evaluation is that 83% of the Redwoods sentences are indeed nets, and 17% aren't. A closer analysis of the non-nets reveals that they seem to be systematically incomplete, i. e. they predict more readings than the sentence actually has. This supports the claim that all linguistically correct MRS expressions are indeed nets. We also verify the other two assumptions, one empirically and one by proof.", "cite_spans": [ { "start": 147, "end": 167, "text": "(Oepen et al., 2002)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Our results are practically relevant because dominance constraint solvers are much faster and have more predictable runtimes when solving nets than the LKB solver for MRS (Copestake, 2002) , as we also show here. In addition, nets might be useful as a debugging tool to identify potentially problematic semantic outputs when designing a grammar.", "cite_spans": [ { "start": 171, "end": 188, "text": "(Copestake, 2002)", "ref_id": "BIBREF7" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Plan of the Paper. We first recall the definitions of MRS ( \u00a72) and dominance constraints ( \u00a73). We present the translation from MRS-nets to dominance constraints ( \u00a74) and prove that it can be extended to MRS-nets with EP-conjunction ( \u00a75). Finally we evaluate the net hypothesis and the qeq assumption on the Redwoods corpus, and compare runtimes ( \u00a76).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "This section presents a definition of Minimal Recursion Semantics (MRS) (Copestake et al., 2004) including EP-conjunctions with a merging semantics. Full MRS with qeq-semantics, top handles, and event variables will be discussed in the last paragraph. MRS Syntax. MRS constraints are conjunctive formulas over the following vocabulary:", "cite_spans": [ { "start": 72, "end": 96, "text": "(Copestake et al., 2004)", "ref_id": "BIBREF6" } ], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "1. An infinite set of variables ranged over by h.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "Variables are also called handles. 2. An infinite set of constants x, y, z denoting indivual variables of the object language. 3. A set of function symbols ranged over by P, and a set of quantifier symbols ranged over by Q.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "Pairs Q x are further function symbols. 4. The binary predicate symbol '= q '. MRS constraints have three kinds of literals, two kinds of elementary predications (EPs) in the first two lines and handle constraints in the third line:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "1. h :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "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 = q h 2 In", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Minimal Recursion Semantics", "sec_num": "2" }, { "text": "EPs, label positions are on the left of ':' and argument positions on the right. Let M be a set of literals. The label set lab(M) contains all handles of M that occur in label but not in argument position, and the argument handle set arg(M) contains all 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": ". An MRS constraint (MRS for short) is a finite set M of MRSliterals such that: M1 every handle occurs at most once in argument position in M, M2 handle constraints h = q h always relate argument handles h to labels h , and 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 constraints)", "sec_num": null }, { "text": "Q x (h 1 , h 2 ) in M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS constraints)", "sec_num": null }, { "text": "We say that an MRS M is compact if every handle h in M is either a label or an argument handle. Compactness simplifies the following proofs, but it is no serious restriction in practice.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS constraints)", "sec_num": null }, { "text": "We usually represent MRSs as directed graphs: the nodes of the graph are the handles of the MRS, EPs are represented as solid lines, and handle constraints are represented as dotted lines. For instance, the following MRS is represented by the graph on the left of Fig. 1 . Note that the relation between bound variables and their binders is made explicit by binding edges drawn as dotted lines (cf. C2 below); transitively redundand binding edges (e. g., from some y to book y ) however are omited. MRS Semantics. Readings of underspecified representations correspond to configurations of MRS constraints. Intuitively, a configuration is an MRS where all handle constraints have been resolved by plugging the \"tree fragments\" into each other.", "cite_spans": [], "ref_spans": [ { "start": 264, "end": 270, "text": "Fig. 1", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Definition 1 (MRS constraints)", "sec_num": null }, { "text": "{h 5 : some y (h 6 , h 8 ), h 7 : book(y), h 1 : every x (h 2 , h 4 ), h 3 : student(x), h 9 : read(x, y), h 2 = q h 3 , h 6 = q h 7 } every x some y student x book y read x,y every x some y student x book y read x,y every x some y student x book y read x,y", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS constraints)", "sec_num": null }, { "text": "Let M be an MRS and h, h be handles in M. We say that h immediately outscopes h in M if there is an EP in M with label h and argument handle h , and we say that h outscopes h in M if the pair (h, h ) belongs to the reflexive transitive closure of the immediate outscope relation of M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 1 (MRS constraints)", "sec_num": null }, { "text": "An MRS M is a configuration if it satisfies conditions C1 and C2: C1 The graph of M is a tree of solid edges: (i) all handles are labels i. e., arg(M) = / 0 and M contains no handle constraints, (ii) handles don't properly outscope themselve, and (iii) all handles are pairwise connected by EPs in M. C2 If h : Q x (h 1 , h 2 ) and h : P (... ,x,...) belong to M, then h outscopes h in M i. e., binding edges in the graph of M are transitively redundant. We say that a configuration M is configuration of an MRS M if there exists a partial substitution \u03c3 : lab(M ) arg(M ) that states how to identify labels with argument handles of M so that:", "cite_spans": [ { "start": 338, "end": 350, "text": "(... ,x,...)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (MRS configurations).", "sec_num": null }, { "text": "C3 M = {\u03c3(E) | E is an EP in M }, and C4 for all h = q h in M , h outscopes \u03c3(h ) in M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (MRS configurations).", "sec_num": null }, { "text": "The value \u03c3(E) is obtained by substituting all labels in dom(\u03c3) in E while leaving all other handels unchanged.", "cite_spans": [ { "start": 57, "end": 63, "text": "dom(\u03c3)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (MRS configurations).", "sec_num": null }, { "text": "The MRS on the left of Fig. 1 , for instance, has two configurations given to the right. EP-conjunctions. Definitions 1 and 2 generalize the idealized definition of MRS of Niehren and Thater (2003) by EP-conjunctions with a merging semantics. An MRS M contains an EP-conjunction if it contains different EPs with the same label h.The intuition is that EP-conjunctions are interpreted by object language conjunctions. We call a configuration merging if it contains EPconjunctions, and merging-free otherwise. Merging configurations are needed to solve EP-conjuctions such as {h : P 1 , h : P 2 }. Unfortunately, they can also solve MRSs without EP-conjunctions, such as the MRS in Fig. 3 . The unique configuration of this MRS is a merging configuration: the labels of P 1 and P 2 must be identified with the only available argument handle. The admission of merging configurations may thus have important consequences for the solution space of arbitrary MRSs.", "cite_spans": [ { "start": 172, "end": 197, "text": "Niehren and Thater (2003)", "ref_id": "BIBREF10" } ], "ref_spans": [ { "start": 23, "end": 29, "text": "Fig. 1", "ref_id": "FIGREF0" }, { "start": 680, "end": 686, "text": "Fig. 3", "ref_id": null } ], "eq_spans": [], "section": "Definition 2 (MRS configurations).", "sec_num": null }, { "text": "P 1 , P 2 P 3 {h 1 : P 1 (h 2 ), h 1 : P 2 (h 3 ), h 4 : P 3 h 2 = q h 4 , h 3 = q h 4 }", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (MRS configurations).", "sec_num": null }, { "text": "Standard MRS requires three further extensions: (i) qeq-semantics, (ii) tophandles, and (iii) event variables. These extensions are less relevant for our comparision.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "The qeq-semantics restricts the interpretation of handle constraints beyond dominance. Let M be an MRS with handles h, h . We say that h is qeq h in M if either h = h , or there is an EP h :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "Q x (h 0 , h 1 ) in M and h 1 is qeq h in M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "Every qeq-configuration is a configuration as defined above, but not necessarily vice versa. The qeq-restriction is relevant in theory but will turn out unproblematic in practice (see \u00a76).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "Standard MRS requires the existence of top handles in all MRS constraints. This condition doesn't matter for MRSs with connected graphs (see (Bodirsky et al., 2004) for the proof idea). MRSs with unconnected graphs clearly do not play any role in practical underspecified semantics.", "cite_spans": [ { "start": 141, "end": 164, "text": "(Bodirsky et al., 2004)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "Finally, MRSs permit events variables e, e as a second form of constants. They are treated equally to individual variables except that they cannot be bound by quantifiers.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Standard MRS.", "sec_num": null }, { "text": "Dominance constraints are a general framework for describing trees. For scope underspecification, they are used to describe the syntax trees of object language formulas. Dominance constraints are the core language underlying CLLS (Egg et al., 2001 ) which adds parallelism and binding constraints.", "cite_spans": [ { "start": 230, "end": 247, "text": "(Egg et al., 2001", "ref_id": "BIBREF8" } ], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "Syntax and semantics. We assume a possibly infinite signature \u03a3 = { f , g,...} of function symbols with fixed arities (written ar( f )) and an infinite set of variables ranged over by X,Y, Z.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "A dominance constraint \u03d5 is a conjunction of dominance, inequality, and labeling literals of the following form, where ar( f ) = n:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "\u03d5 ::= X * Y | X = Y | X : f (X 1 ,... ,X n ) | \u03d5 \u2227 \u03d5", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "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, edgeordered and labeled. A solution for a dominance constraint \u03d5 consists of a tree \u03c4 and an assignment \u03b1 that maps the variables in \u03d5 to nodes of \u03c4 such that all constraints are satisfied: labeling literals X : f (X 1 ,... ,X n ) are satisfied iff \u03b1(X) is labeled with f and its daughters are \u03b1(X 1 ),... ,\u03b1(X n ) in this order; dominance literals X * Y are satisfied iff \u03b1(X) dominates \u03b1(Y ) in \u03c4; and inequality literals X = Y are satisfied iff \u03b1(X) and \u03b1(Y ) are distinct nodes. Solved forms. Satisfiable dominance constraints have infinitely many solutions. Constraint solvers for dominance constraints therefore do not enumerate solutions but solved forms i. e., \"tree shaped\" constraints. To this end, we consider (weakly) normal dominance constraints (Bodirsky et al., 2004) .", "cite_spans": [ { "start": 952, "end": 975, "text": "(Bodirsky et al., 2004)", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "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": "Dominance Constraints", "sec_num": "3" }, { "text": "Definition 3. A dominance constraint \u03d5 is normal 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": "Dominance Constraints", "sec_num": "3" }, { "text": "(b) if X * Y occurs in \u03d5, X is a hole in \u03d5. We call \u03d5 weakly normal if it satisfies the above properties except for N1 (b) and N3 (b).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "Note that Definition 3 imposes compactness: the height of tree fragments is always one. This is not Figure 4 : A normal dominance constraint (left) and its two solved forms (right). a serious restriction, as weakly normal dominance constraints can be compactified, provided that dominance links relate either roots or holes with roots.", "cite_spans": [], "ref_spans": [ { "start": 100, "end": 108, "text": "Figure 4", "ref_id": null } ], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "every x some y student x book y read x,y every x some y student x book y read x,y every x some y student x book y read x,y", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "Weakly normal dominance constraints \u03d5 can be represented by dominance graphs. The dominance graph of \u03d5 is a directed graph G = (V, E T E D ) defined as follows. The nodes of G are the variables of \u03d5. Labeling literals X : f (X 1 ,... ,X k ) are represented by tree edges (X, X i ) \u2208 E T , for 1 \u2264 i \u2264 k, and dominance literals X * X are represented by dominance edges (X, X ) \u2208 E D . Inequality literals are not represented in the graph. In pictures, labeling literals are drawn with solid lines and dominance edges with dotted lines.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "We say that a constraint \u03d5 is in solved form if its graph is in solved form. A graph G is in solved form iff it is a forest. The solved forms of G are solved forms G which are more specific than G i. e., they differ only in their dominance edges and the reachability relation of G extends the reachability of G . A minimal solved form is a solved form which is minimal with respect to specificity. Simple solved forms are solved forms where every hole has exactly one outgoing dominance edge. Fig. 4 shows as a concrete example the translation of the MRS description in Fig. 1 together with its two minimal solved forms. Both solved forms are simple.", "cite_spans": [], "ref_spans": [ { "start": 493, "end": 499, "text": "Fig. 4", "ref_id": null }, { "start": 570, "end": 576, "text": "Fig. 1", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Dominance Constraints", "sec_num": "3" }, { "text": "This section defines MRS-nets without EPconjunctions, and sketches their translation to normal dominance constraints. We define nets equally for MRSs and dominance constraints. The key semantic property of nets is that different notions of solutions coincide. In this section, we show that merging-free configurations coincides to minimal solved forms. \u00a75 generalizes the translation by adding EP-conjunctions and permitting merging semantics. Nets. A hypernormal path (Althaus et al., 2003) in a constraint graph is a path in the undirected graph that contains for every leaf X at most one incident dominance edge. Let \u03d5 be a weakly normal dominance constraint and let G be the constraint graph of \u03d5. We say that \u03d5 is a dominance net if the transitive reduction G of G is a net. G is a net if every tree fragment F of G satisfies one of the following three conditions, illustrated in Fig. 5 : Strong. Every hole of F has exactly one outgoing dominance edge, and there is no weak root-to-root dominance edge. Weak. Every hole except for the last one has exactly one outgoing dominance edge; the last hole has no outgoing dominance edge, and there is exactly one weak root-to-root dominance edge. Island. The fragment has one hole X, and all variables which are connected to X by dominance edges are connected by a hypernormal path in the graph where F has been removed.", "cite_spans": [ { "start": 469, "end": 491, "text": "(Althaus et al., 2003)", "ref_id": "BIBREF1" } ], "ref_spans": [ { "start": 885, "end": 891, "text": "Fig. 5", "ref_id": "FIGREF2" } ], "eq_spans": [], "section": "Translating Merging-Free MRS-Nets", "sec_num": "4" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "h : P(x 1 ,... ,x n , h 1 ,... ,h k ) \u2192 h : P x 1 ,...,x n (h 1 ,... ,h k ) h : Q x (h 1 , h 2 ) \u2192 h : Q x (h 1 , h 2 ) h = q h \u2192 h * h Additionally, dominance literals h * h are added to \u03d5 M for all h, h s. t. h : Q x (h 1 ,", "eq_num": "h" } ], "section": "Pre", "sec_num": null }, { "text": "We say that an MRS M is an MRS-net if the pretranslation of its literals results in a dominance net \u03d5 M . We say that an MRS-net M is connected if \u03d5 M is connected; \u03d5 M is connected if the graph of \u03d5 M is connected.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Pre", "sec_num": null }, { "text": "Note that this notion of MRS-nets implies that MRS-nets cannot contain EP-conjunctions as otherwise the resulting dominance constraint would not be weakly normal. \u00a75 shows that EP-conjunctions can be resolved i. e., MRSs with EP-conjunctions can be mapped to corresponding MRSs without EPconjunctions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Pre", "sec_num": null }, { "text": "If M is an MRS-net (without EP-conjunctions), then M can be translated into a corresponding dominance constraint \u03d5 by first pre-translating M into a \u03d5 M and then normalizing \u03d5 M by replacing weak root-to-root dominance edges in weak fragments by dominance edges which start from the open last hole.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Pre", "sec_num": null }, { "text": "Theorem 1 (Niehren and Thater, 2003) . Let M be an MRS and \u03d5 M be the translation of M. If M is a connected MRS-net, then the merging-free configurations of M bijectively correspond to the minimal solved forms of the \u03d5 M .", "cite_spans": [ { "start": 10, "end": 36, "text": "(Niehren and Thater, 2003)", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "Pre", "sec_num": null }, { "text": "The following section generalizes this result to MRS-nets with a merging semantics.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Pre", "sec_num": null }, { "text": "We now show that if an MRS is a net, then all its configurations are merging-free, which in particular means that the translation can be applied to the more general version of MRS with a merging semantics. Proof. Let M be a configuration of M and let \u03c3 be the underlying substitution. We construct a solved form \u03d5 M as follows: the labeling literals of \u03d5 M are the pre-translations of the EPs in M, and \u03d5 M has a dominance literal h * h iff (h, h ) \u2208 \u03c3, and inequality literals X = Y for all distinct roots in \u03d5 M .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "By condition C1 in Def. 2, the graph of M is a tree, hence the graph of \u03d5 M must also be a tree i. e., \u03d5 M is a solved form. \u03d5 M must also be more specific than the graph of \u03d5 M because the graph of M satisfies all dominance requirements of the handle constraints in M, hence \u03d5 M is a solved form of \u03d5 M .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "M clearly solved \u03d5 M . By Lemmata 2 and 3, \u03d5 M must be simple and minimal because \u03d5 M is a net. But then M cannot contain EP-conjunctions i. e., M is merging-free.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "The merging semantics of MRS is needed to solve EP-conjunctions. As we have seen, the merging semantics is not relevant for MRS constraints which are nets. This also verifies Niehren and Thater's (2003) assumption that EP-conjunctions are \"syntactic sugar\" which can be resolved in a preprocessing step: EP-conjunctions can be resolved by exhaustively applying the following rule which adds new literals to make the implicit conjunction explicit: 2 '(h 1 ,... ,h n , h 1 ,... ,h m ) ,", "cite_spans": [ { "start": 175, "end": 202, "text": "Niehren and Thater's (2003)", "ref_id": "BIBREF10" } ], "ref_spans": [ { "start": 447, "end": 482, "text": "2 '(h 1 ,... ,h n , h 1 ,... ,h m )", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "h : E 1 (h 1 ,... ,h n ), h : E 2 (h 1 ,... ,h m ) \u21d2 h : 'E 1 &E", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "where E(h 1 ,... ,h n ) stands for an EP with argument handles h 1 ,... ,h n , and where 'E 1 &E 2 ' is a complex function symbol. If this rule is applied exhaustively to an MRS M, we obtain an MRS M without EPconjunctions. It should be intuitively clear that the configurations of M and M correspond; Therefore, the configurations of M also correspond to the minimal solved forms of the translation of M .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Merging and EP-Conjunctions", "sec_num": "5" }, { "text": "The two remaining assumptions underlying the translation are the \"net-hypothesis\" that all linguistically relevant MRS expressions are nets, and the \"qeq-hypothesis\" that handle constraints can be given a dominance semantics practice. In this section, we empirically show that both assumptions are met in practice.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Evaluation", "sec_num": "6" }, { "text": "As an interesting side effect, we also compare the run-times of the constraint-solvers we used, and we find that the dominance constraint solver typically outperforms the MRS solver, often by significant margins.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Evaluation", "sec_num": "6" }, { "text": "Grammar and Resources. We use the English Resource Grammar (ERG), a large-scale HPSG grammar, in connection with the LKB system, a grammar development environment for typed feature grammars (Copestake and Flickinger, 2000) . We use the system to parse sentences and output MRS constraints which we then translate into dominance constraints. As a test corpus, we use the Redwoods Treebank (Oepen et al., 2002) which contains 6612 sentences. We exclude the sentences that cannot be parsed due to memory capacities or words and grammatical structures that are not included in the ERG, or which produce ill-formed MRS expressions (typically violating M1) and thus base our evaluation on a corpus containing 6242 sentences. In case of syntactic ambiguity, we only use the first reading output by the LKB system.", "cite_spans": [ { "start": 190, "end": 222, "text": "(Copestake and Flickinger, 2000)", "ref_id": "BIBREF3" }, { "start": 388, "end": 408, "text": "(Oepen et al., 2002)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Evaluation", "sec_num": "6" }, { "text": "To enumerate the solutions of MRS constraints and their translations, we use the MRS solver built into the LKB system and a solver for weakly normal dominance constraints (Bodirsky et al., 2004) , which is implemented in C ++ and uses LEDA, a class library for efficient data types and algorithms (Mehlhorn and N\u00e4her, 1999) .", "cite_spans": [ { "start": 171, "end": 194, "text": "(Bodirsky et al., 2004)", "ref_id": "BIBREF2" }, { "start": 297, "end": 323, "text": "(Mehlhorn and N\u00e4her, 1999)", "ref_id": "BIBREF9" } ], "ref_spans": [], "eq_spans": [], "section": "Evaluation", "sec_num": "6" }, { "text": "We check for 6242 constraints whether they constitute nets. It turns out that 5200 (83.31%) constitute nets while 1042 (16.69%) violate one or more netconditions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Relevant Constraints are Nets", "sec_num": "6.1" }, { "text": "Non-nets. The evaluation shows that the hypothesis that all relevant constraints are nets seems to be falsified: there are constraints that are not nets. However, a closer analysis suggests that these constraints are incomplete and predict more readings than the sentence actually has. This can also be illustrated with the average number of solutions: For the Redwoods corpus in combination with the ERG, nets have 1836 solutions on average, while non-nets have 14039 solutions, which is a factor of 7.7. The large number of solutions for non-nets is due to the \"structural weakness\" of non-nets; often, non-nets have only merging configurations. Non-nets can be classified into two categories (see Fig. 6 ): The first class are violated \"strong\" fragments which have holes without outgoing dominance edge and without a corresponding root-toroot dominance edge. The second class are violated \"island\" fragments where several outgoing dominance edges from one hole lead to nodes which are not hypernormally connected. There are two more possibilities for violated \"weak\" fragmentshaving more than one weak dominance edge or having a weak dominance edge without empty hole-, but they occur infrequently (4.4%). If those weak fragments were normalized, they would constitute violated island fragments, so we count them as such.", "cite_spans": [], "ref_spans": [ { "start": 700, "end": 706, "text": "Fig. 6", "ref_id": "FIGREF5" } ], "eq_spans": [], "section": "Relevant Constraints are Nets", "sec_num": "6.1" }, { "text": "124 (11.9%) of the non-nets contain empty holes, 762 (73.13%) contain violated island fragments, and 156 (14.97%) contain both. Those constraints that contain only empty holes and no violated island fragments cannot be configured, as in configurations, all holes must be filled.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Relevant Constraints are Nets", "sec_num": "6.1" }, { "text": "Fragments with open holes occur frequently, but not in all contexts, for constraints representing for example time specifications (e. g., \"from nine to twelve\" or \"a three o'clock flight\") or intensional expressions (e. g., \"Is it?\" or \"I suppose\" formed island fragments are often triggered by some kind of coordination, like \"a restaurant and/or a sauna\" or \"a hundred and thirty Marks\", also implicit ones like \"one hour thirty minutes\" or \"one thirty\". Constraints with both kinds of violated fragments emerge when there is some input that yields an open hole and another part of the input yields a violated island fragment (for example in constructions like \"from nine to eleven thirty\" or \"the ten o'clock flight Friday or Thursday\", but not necessarily as obviously as in those examples). The constraint on the left in Fig. 7 gives a concrete example for violated island fragments. The topmost fragment has outgoing dominance edges to otherwise unconnected subconstraints \u03d5 1 and \u03d5 2 . Under the merging-free semantics of the MRS dialect used in (Niehren and Thater, 2003) where every hole has to be filled exactly once, this constraint cannot be configured: there is no hole into which \"available\" could be plugged. However, standard MRS has merging configuration where holes can be filled more than once. For the constraint in Fig. 7 this means that \"available\" can be merged in almost everywhere, only restricted by the \"qeq-semantics\" which forbids for instance \"available\" to be merged with \"sauna.\" In fact, the MRS constraint solver derives sixteen configurations for the constraint, two of which are given in Fig. 7 , although the sentence has only two scope readings.", "cite_spans": [ { "start": 1053, "end": 1079, "text": "(Niehren and Thater, 2003)", "ref_id": "BIBREF10" } ], "ref_spans": [ { "start": 826, "end": 832, "text": "Fig. 7", "ref_id": null }, { "start": 1336, "end": 1342, "text": "Fig. 7", "ref_id": null }, { "start": 1624, "end": 1630, "text": "Fig. 7", "ref_id": null } ], "eq_spans": [], "section": "Relevant Constraints are Nets", "sec_num": "6.1" }, { "text": "We conjecture that non-nets are semantically \"incomplete\" in the sense that certain constraints are missing. For instance, an alternative analysis for the above constraint is given in Fig. 8 . The constraint adds an additional argument handle to \"and\" and places a dominance edge from this handle to \"available.\" In fact, the constraint is a net; it has exactly two readings.", "cite_spans": [], "ref_spans": [ { "start": 184, "end": 190, "text": "Fig. 8", "ref_id": null } ], "eq_spans": [], "section": "Relevant Constraints are Nets", "sec_num": "6.1" }, { "text": "For all nets, the dominance constraint solver calculates the same number of solutions as the MRS solver does, with 3 exceptions that hint at problems in the syntax-semantics interface. As every configuration that satisfies proper qeq-constraints is also a configuration if handle constraints are interpreted under the weaker notion of dominance, the solutions computed by the dominance constraint solver and the MRS solver must be identical for every constraint. This means that the additional expressivity of proper qeq-constraints is not used in practice, which in turn means that in practice, the translation is sound and correct even for the standard MRS notion of solution, given the constraint is a net.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Qeq is dominance", "sec_num": "6.2" }, { "text": "The availability of a large body of underspecified descriptions both in MRS and in dominance constraint format makes it possible to compare the solvers for the two underspecification formalisms. We measured the runtimes on all nets using a Pentium III CPU at 1.3 GHz. The tests were run in a multi-user environment, but as the MRS and dominance measurements were conducted pairwise, conditions were equal for every MRS constraint and corresponding dominance constraint.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Comparison of Runtimes", "sec_num": "6.3" }, { "text": "The measurements for all MRS-nets with less than thirty dominance edges are plotted in Fig. 9 . Inputs are grouped according to the constraint size. The filled circles indicate average runtimes within each size group for enumerating all solutions using the dominance solver, and the empty circles indicate the same for the LKB solver. The brackets around each point indicate maximum and minimum runtimes in that group. Note that the vertical axis is logarithmic.", "cite_spans": [], "ref_spans": [ { "start": 87, "end": 93, "text": "Fig. 9", "ref_id": "FIGREF7" } ], "eq_spans": [], "section": "Comparison of Runtimes", "sec_num": "6.3" }, { "text": "We excluded cases in which one or both of the solvers did not return any results: There were 173 sentences (3.33% of all nets) on which the LKB solver ran out of memory, and 1 sentence (0.02%) that took the dominance solver more than two minutes to solve.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Comparison of Runtimes", "sec_num": "6.3" }, { "text": "The graph shows that the dominance constraint solver is generally much faster than the LKB solver: The average runtime is less by a factor of 50 for constraints of size 10, and this grows to a factor of 500 for constraints of size 25. Our experiments show that the dominance solver outperforms the LKB solver on 98% the cases. In addition, its runtimes are much more predictable, as the brackets in the graph are also shorter by two or three orders of magnitude, and the standard deviation is much smaller (not shown).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Comparison of Runtimes", "sec_num": "6.3" }, { "text": "We developed Niehren and Thater's (2003) theoretical translation into a practical system for translating MRS into dominance constraints, applied it systematically to MRSs produced by English Resource Grammar for the Redwoods treebank, and evaluated the results. We showed that:", "cite_spans": [ { "start": 13, "end": 40, "text": "Niehren and Thater's (2003)", "ref_id": "BIBREF10" } ], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "1. most \"real life\" MRS expressions are MRSnets, which means that the translation is correct in these cases; 2. for nets, merging is not necessary (or even possible); 3. the practical translation works perfectly for all MRS-nets from the corpus; in particular, the = q relation can be taken as synonymous with dominance in practice. Because the translation works so well in practice, we were able to compare the runtimes of MRS and dominance constraint solvers on the same inputs. This evaluation shows that the dominance constraint solver outperforms the MRS solver and displays more predictable runtimes. A researcher working with MRS can now solve MRS nets using the efficient dominance constraint solvers.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "A small but significant number of the MRS constraints derived by the ERG are not nets. We have argued that these constraints seem to be systematically incomplete, and their correct completions are indeed nets. A more detailed evaluation is an important task for future research, but if our \"net hypothesis\" is true, a system that tests whether all outputs of a grammar are nets (or a formal \"safety criterion\" that would prove this theoretically) could be a useful tool for developing and debugging grammars.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "From a more abstract point of view, our evaluation contributes to the fundamental question of what expressive power an underspecification formalism needs. It turned out that the distinction between qeq and dominance hardly plays a role in practice. If the net hypothesis is true, it also follows that merging is not necessary because EP-conjunctions can be converted into ordinary conjunctions. More research along these lines could help unify different underspecification formalisms and the resources that are available for them.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" } ], "back_matter": [ { "text": "We are grateful to Ann Copestake for many fruitful discussions, and to our reviewers for helpful comments.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Acknowledgments", "sec_num": null } ], "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 se- mantic interpretation. 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The function symbols of both EPs are conjoined and their arguments are merged into a set. The MRS does not have configurations since the argument handles of the merged EPs cannot jointly outscope the node P 4 ." }, "FIGREF2": { "type_str": "figure", "uris": null, "num": null, "text": "-translation. An MRS constraint M can be represented as a corresponding dominance constraint \u03d5 M as follows: The variables of \u03d5 M are the handles of M, and the literals of \u03d5 M correspond Fragment Schemata of Nets those of M in the following sence:" }, "FIGREF3": { "type_str": "figure", "uris": null, "num": null, "text": "2 ) and h : P(... ,x,...) belong to M (cf. C2), and literals h = h are added to \u03d5 M for all h, h in distinct label position in M.Lemma 1. If a compact MRS M does not contain EP-conjunctions then \u03d5 M is weakly normal, and the graph of M is the transitive reduction of the graph of \u03d5 M ." }, "FIGREF4": { "type_str": "figure", "uris": null, "num": null, "text": "Niehren and Thater, 2003). All minimal solved forms of a connected dominance net are simple.Lemma 3. If all solved forms of a normal dominance constraint are simple, then all of its solved forms are minimal. Theorem 2. The configurations of an MRS-net M are merging-free." }, "FIGREF5": { "type_str": "figure", "uris": null, "num": null, "text": "Two classes of non-nets" }, "FIGREF6": { "type_str": "figure", "uris": null, "num": null, "text": "An MRS for \"A sauna and a cafeteria are available\" (top) and two of sixteen merging configurations (below). The \"repaired\" MRS fromFig. 7" }, "FIGREF7": { "type_str": "figure", "uris": null, "num": null, "text": "Comparison of runtimes for the MRS and dominance constraint solvers." } } } }