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Markus Egg, Alexander Koller, Joachim Niehren The Constraint Language for Lambda Structures

Markus Egg, Alexander Koller, Joachim Niehren The Constraint Language for Lambda Structures. Ventsislav Zhechev SfS, Universität Tübingen e-mail: vzhechev@sfs.uni-tuebingen.de. Introduction Motivation Basic Terms. Elements of CLLS -terms -structures Discussed Phenomena. Agenda.

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Markus Egg, Alexander Koller, Joachim Niehren The Constraint Language for Lambda Structures

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  1. Markus Egg, Alexander Koller, Joachim NiehrenThe Constraint Language for Lambda Structures Ventsislav Zhechev SfS, Universität Tübingen e-mail: vzhechev@sfs.uni-tuebingen.de

  2. Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena Agenda • Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena

  3. Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena Agenda (continued) • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs

  4. Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena Agenda (continued) • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis

  5. Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena Agenda (continued) • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion

  6. Introduction • Motivation • Basic Terms Introduction Motivation Basic Terms • Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena • Linguistic Phenomena • Scope Ambiguities • Anaphora • VP Ellipsis • Underspecification Introduction Motivation

  7. Introduction Motivation Basic Terms • Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena • Trees • Underspecification • Constraint Language for Lambda Structures:A Combination of Constraints • Dominance Constraints • Anaphoric Binding Constraints • Parallelism Constraints • -binding Constraints Basic Terms

  8. Introduction • Motivation • Basic Terms • Elements of CLLS • -terms • -structures • Discussed Phenomena • Elements of CLLS • -terms • -structures • Discussed Phenomena • Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Every linguist attends a workshop. • (a workshop)(x (every linguist)(y (attend x) y)) • Types • e  individuals • t  truth values (0 or 1 / true or false) • <e,t>  one-place predicates • <e,<e,t>> two-place predicates • etc. Elements of CLLS -terms

  9. Elements of CLLS • -terms • -structures • Discussed Phenomena • Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs attend • (a workshop)(x (every linguist)(y (attend x) y)) • lam  -abstraction • @  functional application • var  bound variable •  variable binding -structures

  10. Elements of CLLS • -terms • -structures • Discussed Phenomena • Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Scope Ambiguity • Every linguist attends a workshop. • (a workshop)(x (every linguist)(y (attend x) y)) • (every linguist)(y (a workshop)(x (attend x) y)) •  dominance Discussed Phenomena

  11. Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • VP Ellipsis • Every man sleeps, and so does Mary. • Parallelism Constraint:X1/X2~Y1/Y2

  12. Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Anaphora • Johni said heij liked hisjmother. • ana  anaphora •  anaphoric link

  13. Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • The Capturing Problem • Variable binding in -terms is usually indicated by using variable names, i.e. x binds all occurrences of x in its scope • Possible Problems: • -calculus has to exclude the capturing of free variables by unintended binders • Problems with constraints used for scope ambiguities • Problems in the presence of parallelism constraints

  14. Elements of CLLS • -terms • -structures • Discussed Phenomena • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Trees and Tree Structures • The Algebra of Trees • Let  be a set of function symbols, f, g, a, b • Each function symbol f has fixed arity • We write fk for a function symbol f with arity k ≥ 0 • We define tree as a ground term built from a set of function symbols • We define path as a word over ℕ(the natural numbers) • We identify each node in a tree with the path from the root to this node • The empty word, , identifies the root • Concatenation is written as  • A word  is a prefix of , iff there is a word 1such that =1 Syntax and Semantics of CLLS Tree Structures

  15. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Tree Structures • tree domain  is a finite nonempty set of nodes, which is prefix closed (  ) and closed under left siblings (i  j for all 1≤j<i) • tree structure is defined as follows: • Given nodes 0,..., n, we write 0:f(1,..., n) for(0, 1,..., n)  :f

  16. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Tree Structures and -structures • Formalization • For -structures we assume:{var0, ana0, lam1, @2}   • We define -structures as follows: • We draw -structures as tree-like graphs -structures

  17. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Dominance • Let  be a -structure and ,  two of its nodes.We say that  dominates  (⊲*), if  lies above ,i.e.  is a prefix of  • Dominance is a partial order on the domain of  and it is reflexive, transitive and antisymmetric • Parallelism • We call any pair / of nodes ,  in  with ⊲* a segment of , where  is called the root and  the hole of the segment • We define: Dominance and Parallelism

  18. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Correspondence Functions between segments: • Parallelism Relation:

  19. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis

  20. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • We assume an infinite set of node variables, ranged over X, Xi, Y, etc. • We pick relation symbols for all relations defined so far • Finally we define CLLS with the following abstract syntax: • The Semantics of CLLS is defined by interpretation of constraints over the class of-structures: • A pair of a -structure  and a variable assignment  into the domain of  satisfies a constraint , iff it satisfies each atomic conjunct of it • We call (, ) a solution of  in this case The CLLS

  21. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • CLLS Constraints are usually hard to read in the standard syntax. That is why we will use constraint graphs for presenting the constraints • For Example: Constraint Graphs

  22. Syntax and Semantics of CLLS • Tree Structures • -structures • Dominance and Parallelism • The CLLS • Constraint Graphs • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Throughout the chapter we assume a fixed signature:={@2, lam1, var0, ana0, before2, mary0, read0, ...} • We follow the convention that proper nouns are always analyzed as constants of type e, except as contrasting elements in ellipses where the other contrasting element is a quantifier Interaction of Quantifiers, Anaphora and Ellipsis

  23. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Every linguist attends a workshop. • Every computer scientist does, too. • The pair of sentences has three possible readings, although it may seem that there are four • The CLLS constraint for the two sentences looks like this: Quantifier Parallelism

  24. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • John likes his mother, and Bill does too. • The sentence has two readings: strict (Bill likes John’s mother) and sloppy (Bill likes Bill’s mother) • We describe the meaning of the sentence using parallelism and anaphoric linking constraints: Strict/Sloppy Ambiguities

  25. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • According to the parallelism constraint the tree part of the -structure below Xt is the same as the one below Xs, except for the contrasting elements, as follows: • This is yet not a complete -structure, because the anaphor at Xa’ doesn’t have an antecedent

  26. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • John revised his paper before the teacher did, and so did Bill. • This sentence comprises nested ellipsis: the source clause of the ellipsis is elliptical itself • The sentence is further complicated by the presence of the anaphor, which induces a complex strict/sloppy ambiguity • We follow Dalrymple et al. (1991) in assuming five readings for the sentence Nested Ellipses

  27. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • All five readings are represented by the following constraint:

  28. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Mary read a book she liked before Sue did. • The sentence has three readings • In the first reading the indefinite NP a book she liked outscopes both clauses • The second and the third reading arise from a strict/sloppy ambiguity that occurs if the operator before outscopes the indefinite • Here is a constraint describing the readings: A Complex Interaction

  29. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Schematic representations of the solutions: • First reading: • Second reading: • Third reading:

  30. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • John greeted every person that Max did. • The problem is that the ellipsis is contained in the VP it refers to • In CLLS the meaning of the sentence is described as follows: Antecedent-Contained Ellipsis

  31. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • There is one problem with this analysis:the notion of binding equivalence as defined is too strong a restriction for ACD • The redefinition is given as:

  32. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • The preconditions for the two branches of the definitions are given here as (a) and (b) respectively: • This analysis also accounts for the difference between the following two sentences (the first one lacking one of the two readings of the second sentence): • John wants Bill to read everything that Max does. • John wants Bill to read everything Max wants him to read.

  33. Interaction of Quantifiers, Anaphora and Ellipsis • Quantifier Parallelism • Strict/Sloppy Ambiguities • Nested Ellipses • A Complex Interaction • Antecedent-Contained Ellipsis • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Computational Aspects • Conclusion • Phrase Structure Rules: • The Lexicon is defined by a relation Lex, which relates words W and lexical categories{Det, N, IV, TV, SV, RP, ...}. Terminal productions (a13) expand lexical categories to words of this category The Syntax-Semantics Interface Grammar

  34. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • The Syntax-Semantics Interface should factor out as much of the constraint construction as possible into the interface rules • Most of the lexical entries introduce just one labeling constraint • For each node in the syntax tree a constraint is generated; the constraint of the whole tree is the conjunction of these subconstraints • Each node ℕ* in the syntax tree is associated with two variables, Xs (the local scope domain of ) and Xr (the root of the subconstraint for ) Semantic Construction

  35. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • We add a constraint Xs⊲*Xr for each determiner  which is not an indefinite. We also add this constraint whenever  is a verb • We associate with each NP an index i that is used in the syntactic tree for coindexation with a variable Xi • The variables associated with syntactic nodes are related by the following rules:

  36. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion

  37. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion

  38. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion

  39. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • The complete constraint which the interface produces is the conjunction of all the local constraints we just mentioned, plus the labeling constraints for the lexical entries, of the typeXr:sleep • Exceptions to this rule: • The elliptic does (too) does not add a labeling constraint; its semantics is determined via a parallelism constraint • Whenever coindexation signifies a relation between an anaphor ’ and its antecedent , we add the constraintXr=Xi, when we process  and the constraint X’r:ana ante(X’r)=Xi when we process ’

  40. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • The constraint for a relative pronoun with index i at  isXr=Xi  Xi:var; and the constraint for the corresponding trace (say, at ’) is X’r=Xi. This, together with rule (b11), enforces correct binding of the trace • The constraints for possessive pronouns, such as his, are as follows:

  41. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Every linguist attends a workshop. • First the lexical elements introduce several labeling constraints:X11r:every, X121r:linguist, X21r:attend,X221r:a, X2221r:workshop An Example

  42. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion A1 A2 L1 L2 • The constraints for the NPs are built from the above by the rules (b8) and (b9): • A1:@(XA2r, XL1r)  A2:@(X11r, X121r)  L1:lam(XL2r) X1r:var  XL2r⊲*X1r  (X1r)=XL1r  XL1r≠X1r  X1s=X11s=X121s • A3:@(XA4r, XL3r)  A4:@(X221r, X2221r)  L3:lam(XL4r) X22r:var  XL42⊲*X22r  (X22r)=XL3r  XL3r≠X22r X22s=X221s=X2221s • Then rule (b3) combines the transitive verb and its object • X2r:@(X21r, X22r)  X2s=X21s=X22s

  43. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion A3 A1 A4 A2 L3 L1 L4 L2 Xr X2r • Rule (b1) analogously combines the subject and the VP • Xr:@(X2r, X1r)  Xs=X2s=X1s • So far we have the following constraint: • X11r:every  X121r:linguist  X21r:attend  X221r:a X2221r:workshop  A1:@(XA2r, XL1r)  A2:@(X11r, X121r) L1:lam(XL2r)  X1r:var  XL2r⊲*X1r  (X1r)=XL1r  XL1r≠X1r A3:@(XA4r, XL3r)  A4:@(X221r, X2221r)  L3:lam(XL4r)  X22r:var XL42⊲*X22r  (X22r)=XL3r  XL3r≠X22r  X2r:@(X21r, X22r) Xr:@(X2r, X1r)  X1s=X11s=X121s=X22s=X221s=X2221s=X21s=Xs=X2s

  44. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion • Finally we add the relevant scope island constraints: • The complete sentence is associated with the variable Xs, and all other Xs variables are forced to be equal to this one by other constraints • Node 11 is a determiner and node 21 is a verb; so we add the constraint X11s ⊲* X11r  X21s ⊲* X21r

  45. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Conclusion A3 A1 A2 A4 L3 L1 L4 X121r L2 X221r X2221r Xr X2r X1r X22r • The final constraint we get is the following: • X11r:every  X121r:linguist  X21r:attend  X221r:a X2221r:workshop  A1:@(XA2r, XL1r)  A2:@(X11r, X121r) L1:lam(XL2r)  X1r:var  XL2r⊲*X1r  (X1r)=XL1r  XL1r≠X1r A3:@(XA4r, XL3r)  A4:@(X221r, X2221r)  L3:lam(XL4r)  X22r:var XL42⊲*X22r  (X22r)=XL3r  XL3r≠X22r  X2r:@(X21r, X22r) Xr:@(X2r, X1r)  X1s=X11s=X121s=X22s=X221s=X2221s=X21s=Xs=X2s Xs ⊲* X11r  Xs ⊲* X21r

  46. The Syntax-Semantics Interface • Grammar • Semantic Construction • An Example • Computational Aspects • Computational Aspects • Conclusion • Conclusion • Disambiguation of arbitrary CLLS description is very complex: it has been shown that even CLLS without binding is equivalent to context unification, whose decidability is an open problem in theoretical computer science • There are, however, semi-decision procedures which will eventually enumerate all solved forms of a constraint • For the sublanguage of dominance constraints it was shown, that the satisfiability problem is decidable, but NP-complete Computational Aspects

  47. Computational Aspects • Conclusion • An implementation of a solver for dominance constraints can be obtained by employing constraint programming with finite sets. The constraints can be solved by always performing deterministic propagation steps to eliminate hopeless choices before making case distinctions. • It can be shown that all dominance constraints that are needed for the linguistic application belong to a fragment called normal dominance constraints. Satisfiability of a normal constraint can be checked by a graph algorithm of polynomial runtime; each reading can be enumerated in polynomial time as well. However the graph algorithm is not a complete solver for all dominance constraints.

  48. Computational Aspects • Conclusion • Conclusion • CLLS allows the representation of scope ambiguities, anaphora and ellipsis in simple underspecified structures that are transparent and suitable for processing. • We have shown that CLLS correctly represents many notorious problems from the literature involving scope, anaphora, ellipses and their interactions. • Furthermore CLLS can be used to model reinterpretation (meaning shift) of aspect and NPs in an underspecified way. • Nevertheless the linguistic coverage of CLLS still has to be extended. • Various more formal aspects can also be pursued in the future. Conclusion

  49. Conclusion Thank you!

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