Constraints over Lambda-Structures in Semantic Underspecification

Constraints over Lambda-Structures in Semantic Underspecification Markus Egg and Joachim Niehren* and Peter Ruhrberg and Feiyu Xu Department of Computational Linguistics / *Programming Systems Lab Universit/~t des Saarlandes, Saarbrficken, Germany {egg, peru, feiyu}@coli, uni-sb, de niehren0ps, uni-sb, de Abstract We introduce a first-order language for seman- tic underspecification that we call Constraint Language for Lambda-Structures (CLLS). A A- structure can be considered as a A-term up to consistent renaming of bound variables (a- equality); a constraint of CLLS is an underspec- ified description of a A-structure. CLLS solves a capturing problem omnipresent in underspec- ified scope representations. CLLS features con- straints for dominance, lambda binding, paral- lelism, and anaphoric links. Based on CLLS we present a simple, integrated, amt underspecified treatment of scope, parallelism, and anaphora. 1 Introduction A central concern of semantic underspecifica- tion (van Deemter and Peters, 1996) is the un- derspecification of the scope of variable bind- ing operators such as quantifiers (Hobbs and Shieber, 1987; Alshawi, 1990; Reylc, 1993). This immediately raises the conceptual problem of how to avoid variable-capturing when instan- tiating underspecified scope representations. In principle, capturing may occur in all formalisms for structural underspecification which repre- sent binding relations by the coordination of variables (Reyle, 1995; Pinkal, 1996; Bos, 1996; Niehren et al., 1997a). Consider for instance the verb phrase in (1) Manfred [w, knows every student] An underspecified description of the composi- tional semantics of the VP in (1) might be given along the lines of (2): (2) X=Cl(Vx(student(x)~C2(know(Z, x)))) The recta-variable X in (2) denotes some tree representing a predicate logic formula which is underspecified for quantifier scope by means of two place holders C1 and Cs where a subject- quantifier can be filled in, and a place holder Z for the subject-variable. The binding of the object-variable x by the object-quantifier Vx is coordinated through the name of the object- variable, namely 'x'. Capturing occurs when a new quantifier like 3x is filled in C2 whereby the binding between x and Vx is accidentally undone, and is replaced with a binding of x by ~x. Capturing problems raised by variable coordi- nation may be circumvented in simple cases where all quantifiers in underspecified descrip- tions can be assumed to be named by distinct variables. However, this assumption becomes problematic in the light of parallelism between the interpretations of two clauses. Consider for instance the correction of (1) in (3): (3) No, Hans [vv knows every student] The description of the semantics of the VP in (3) is given in (4): (4) Y=C3(Vy(student(y)~C4(know(Z', y)))) But a full understanding of the combined clauses (1) and (3) requires a grasp of the se- mantic identity of the two VP interpretations. Now, the VP interpretations (2) and (4) look very much alike but for the different object- variable, namely 'y' instead of 'x'. This illus- trates that in cases of parallelism, like in cor- rections, different variables in parallel quanti- fled structures have to be matched against each other, which requires some form of …

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