<?xml version="1.0" standalone="yes"?>
<Paper uid="P98-1058">
  <Title>Constraints over Lambda-Structures in Semantic Underspecification</Title>
  <Section position="2" start_page="0" end_page="353" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> A central concern of semantic underspecification (van Deemter and Peters, 1996) is the underspecification of the scope of variable binding operators such as quantifiers (Hobbs and Shieber, 1987; Alshawi, 1990; Reyle, 1993).</Paragraph>
    <Paragraph position="1"> This immediately raises the conceptual problem of how to avoid variable-capturing when instantiating underspecified scope representations. In principle, capturing may occur in all formalisms for structural underspecification which represent 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 \[vF knows every student\] An underspecified description of the compositional 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 meta-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 C2 where a subjectquantifier 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 objectvariable, 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 3x.</Paragraph>
    <Paragraph position="2"> Capturing problems raised by variable coordination may be circumvented in simple cases where all quantifiers in underspecified descriptions 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 \[vP knows every student\] 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 semantic identity of the two VP interpretations. Now, the VP interpretations (2) and (4) look very much Mike but for the different objectvariable, namely 'y' instead of 'x'. This illustrates that in cases of parallelism, like in corrections, different variables in parallel quantified structures have to be matched against each other, which requires some form of renaming to be done on them. While this is unproblematic for fully specified structures, it presents serious problems with underspecified structures like (2) and (4), as there the names of the vari- null ables are crucial for insuring the right bindings. Any attempt to integrate parallelism with scope underspecification thus has to cope with conflicting requirements on the choice of variable names. Avoiding capturing requires variables to be renamed apart but parallelism needs parallel bound variables to be named alike.</Paragraph>
    <Paragraph position="3"> We avoid all capturing and renaming problems by introducing the notion of A-structures, which represent binding relations without naming variables. A A-structure is a standard predicate logic tree structure which can be considered as a A-term or some other logical formula up-to consistent renaming of bound variables (a-equality). Instead of variable names, a A-structure provides a partial function on tree-nodes for expressing variable binding. An graphical illustration of the A-structure corresponding to the A-term Ax.like(x,x) is given (5). (5) ( ', Axlike(x,x) Formally, the binding relation of the A-structure in (5) is expressed through the partial function A (5) defined by A(5)(v2) = v0 and A(5)(v3) = v0. We propose a first-order constraint language for A-structures called CLLS which solves the capturing problem of underspecified scope representations in a simple and elegant way. CLLS subsumes dominance constraints (Backofen et al., 1995) as known from syntactic processing (Marcus et al., 1983) with tree-adjoining grammars (Vijay-Shanker, 1992; Rogers and Vijay-Shanker, 1994). Most importantly, CLLS constraints can describe the binding relation of a A-structure in an underspecified manner (in contrast to A-structures like (5), which are always fully specified). The idea is that A-binding behaves like a kind of rubber band that can be arbitraryly enlarged but never broken. E.g., (6) is an underspecified CLLS-description of the A-structure (5).</Paragraph>
    <Paragraph position="5"> The constraint (6) does not determine a unique A-structure since it leaves e.g. the space between the nodes X2 and X3 underspecified.</Paragraph>
    <Paragraph position="6"> Thus, (6) may eventually be extended, say, to a constraint that fully specifies the A-structure for the A-term in (7).</Paragraph>
    <Paragraph position="7"> (7) Ay.Az.and(person(y), like(y, z) ) Az intervenes between Ay and an occurrence of y when extending (6) to a representation of (7) without the danger of undoing their binding.</Paragraph>
    <Paragraph position="8"> CLLS is sufficiently expressive for an integrated treatment of semantic underspecification, parallelism, and anaphora. To this purpose it provides parallelism constraints (Niehren and Koller, 1998) of the form X/X',,~Y/Y I reminiscent to equality up-to constraints (Niehren et al., 1997a), and anaphoric bindings constraints of the form ante(X)=X'.</Paragraph>
    <Paragraph position="9"> As proved in (Niehren and Koller, 1998), CLLS extends the expressiveness of context unification (Niehren et al., 1997a). It also extends its linguistic coverage (Niehren et al., 1997b) by integrating an analysis of VP ellipses with anaphora as in (Kehler, 1995). Thus, the coverage of CLLS is comparable to Crouch (1995) and Shieber et al. (1996). We illustrate CLLS at a benchmark case for the interaction of scope, anaphora, and ellipsis (8).</Paragraph>
    <Paragraph position="10"> (8) Mary read a book she liked before Sue did.</Paragraph>
    <Paragraph position="11"> The paper is organized as follows. First, we introduce CLLS in detail and define its syntax and semantics. We illustrate CLLS in sec. 3 by applying it to the example (8) and compare it to related work in the last section.</Paragraph>
  </Section>
class="xml-element"></Paper>