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<Paper uid="P98-1058">
  <Title>Constraints over Lambda-Structures in Semantic Underspecification</Title>
  <Section position="3" start_page="353" end_page="356" type="metho">
    <SectionTitle>
2 A Constraint Language for
A-Structures (CLLS)
</SectionTitle>
    <Paragraph position="0"> CLLS is an ordinary first-order language interpreted over A-structures. A-structures are particular predicate logic tree structures we will introduce. We first exemplify the expressiveness of CLLS.</Paragraph>
    <Section position="1" start_page="353" end_page="355" type="sub_section">
      <SectionTitle>
2.1 Elements of CLLS
</SectionTitle>
      <Paragraph position="0"> A A-structure is a tree structure extended by two additional relations (the binding and the linking relation). We represent A-structures as graphs. Every A-structure characterizes a unique A-term or a logical formula up to consistent renaming of bound variables (a-equality). E.g., the A-structure (10) characterizes the higher-order logic (HOL) formula (9).</Paragraph>
      <Paragraph position="2"> Two things are important here: the label '~' represents explicitly the operation of function application, and the binding of the variable x by the A-operator Ax is represented by an explicit binding relation A between two nodes, labelled as var and lain. As the binding relation is explicit, the variable and the binder need not be given a name or index such as x.</Paragraph>
      <Paragraph position="3"> We can fully describe the above A-structure by means of the constraints for immediate dominance and labeling X:f(X1,..., Xn), (e.g.</Paragraph>
      <Paragraph position="4"> X1:@(X2,)(3) and X3:lam(X4) etc.) and binding constraints A(X)=Y. It is convenient to display such constraints graphically, in the style of (6). The difference of graphs as constraints and graphs as A-structures is important since under-specified structures are always seen as descriptions of the A-structures that satisfy them* Dominance. As a means to underspecify Astructures, CLLS employs constraints for dominance X~*Y. Dominance is defined as the transitive and reflexive closure of immediate dominance. We represent dominance constraints graphically as dotted lines. E.g., in (11) we have the typical case of undetermined scope. It is analysed by constraint (12), where two nodes X1 and X2, lie between an upper bound Xo and a lower bound X3. The graph can be linearized by adding either a constraint XI~*X2 or X2~*X1, resulting in the two possible scoping readings for the sentence (11).</Paragraph>
      <Paragraph position="5">  (11) Every linguist speaks two Asian languages.</Paragraph>
      <Paragraph position="6"> (12) ..... &amp;quot;.X.o.</Paragraph>
      <Paragraph position="8"> Parallelism. (11) may be continued by an elliptical sentence, as in (13).</Paragraph>
      <Paragraph position="9"> (13) Two European ones too.</Paragraph>
      <Paragraph position="10"> We analyse elliptical constructions by means of a parallelism constraint of the form (14) X,/Xp~YdY p which has the intuitive meaning that the semantics Xs of the source clause (12) is parallel to the semantics Yt of the elliptical target clause, up-to the exceptions Xp and Yp, which are the semantic representations of the so called parallel elements in source and target clause. In this case the parallel elements are the two subject NPs.</Paragraph>
      <Paragraph position="11"> (11) and (13) together give us a 'Hirschbiihler sentence' (Hirschbiihler, 1982), and our treatment in this case is descriptively equivalent to that of (Niehren et al., 1997b). Our parallelism constraints and their equality up-to constraints have been shown to be (non-trivially) intertranslatable (Niehren and Koller, 1998) if binding and linking relations in A-structures are ignored.</Paragraph>
      <Paragraph position="12"> For the interaction of binding with parallelism we follow the basic idea that binding relations should be isomorphic between two similar substructures. The cases where anaphora interact with ellipsis are discussed below.</Paragraph>
      <Paragraph position="13"> Anaphoric links. We represent anaphoric dependencies in A-structures by another explicit relation between nodes, the linking relation. An anaphor (i.e. a node labelled as ana) may be linked to an antecedent node, which may be labelled by a name or var, or even be another anaphor. Thus, links can form chains as in (15), where a constraint such as ante(X3)=X2 is represented by a dashed line from X3 to X2.</Paragraph>
      <Paragraph position="14"> The constraint (15) analyzes (16), where the second pronoun is regarded as to be linked to the first, rather than linked to the proper name:  In a semantic interpretation of A-structures, analoguously to a semantics for lambda terms, 1 linked nodes get identical denotations. Intuitively, this means they are interpreted as if names, or variables with their binding relations, would be copied down the link chain. It is crucial though not to use such copied structures right away: the link relation gives precise control over strict and sloppy interpretations when anaphors interact with parallelism.</Paragraph>
      <Paragraph position="15"> E.g., (16) is the source clause of the manypronouns-puzzle, a problematic case of interaction of ellipsis and anaphora. (Xu, 1998), where our treatment of ellipsis and anaphora was developed, argues that link chains yield the best explanation for the distribution of strict/sloppy readings involving many pronouns.</Paragraph>
      <Paragraph position="16"> The basic idea is that an elided pronoun can either be linked to its parallel pronoun in the source clause (referential parallelism) or be linked in a structurally parallel way (structural parallelism). This analysis agrees with the proposal made in (Kehler, 1993; Kehler, 1995). It covers a series of problematic cases in the literature such as the many-pronouns-puzzle, cascaded ellipsis, or the five-reading sentence (17): (17) John revised his paper before the teacher did, and so did Bill The precise interaction of parallelism with binding and linking relations is spelled out in sec. 2.2.</Paragraph>
    </Section>
    <Section position="2" start_page="355" end_page="356" type="sub_section">
      <SectionTitle>
2.2 Syntax and Semantics of CLLS
</SectionTitle>
      <Paragraph position="0"> We start with a set of labels E= {@2, lam I ' var 0 ' ana 0 ' before 2, maryO, readO,,, .}, ranged over by \]ji, with arity i which may be omitted. The syntax of CLLS is given by:</Paragraph>
      <Paragraph position="2"> The semantics of CLLS is given in terms of first order structures L, obtained from underlying tree structures, by adding relations eL for each CLLS relation symbol C/ E {~*, A(.)= &amp;quot;, ante(.)=., ./.~-/-, :@, :lam, :vat,...}. 1We abstain from giving such a semantics here, as we would have to introduce types, which are of no concern here, to keep the semantics simple.</Paragraph>
      <Paragraph position="3"> A (finite) tree structure, underlying L, is given by a set of nodes u, u', ... connected by paths ~r, ~ff, ... (possibly empty words over positive integers), and a labelling \]junction I from nodes to labels. The number of daughters of a node matches the arity of its label. The relationship</Paragraph>
      <Paragraph position="5"> reached from v by following the path 7r (if defined). To express that a path lr is defined on a node v in L we write v.rSL. We write ~r&lt;r' for ~r being an initial segment of 7d. The dominance relation v&lt;~v' holds if 37r v.Tr = v'. If ~r is non-empty we have proper dominance v&lt;+v '.</Paragraph>
      <Paragraph position="6"> A A-structure L is a tree structure with two (partially functional) binary relations AL(')= &amp;quot;, for binding, and anteL(')=', for anaphor-toantecedent linking. We assume that the following conditions hold: (1) binding only holds between variables (nodes labelled var) to A-binders (nodes labelled lain); (2) every variable has exactly one binder; (3) variables are dominated by their binders; (4) only anaphors (nodel labelled ana) are linked to antecendents; (2) every anaphor has exactly one antecendent; (5) antecedents are terminal nodes; (6) there are no cyclic link chains; (7) if a link chain ends at a variable then each anaphor in the chain must be dominated by the binder of that variable.</Paragraph>
      <Paragraph position="7"> The not so straight forward part of the semantics of CLLS is the notion of parallelism, which we define for any given A-structure L as follows: iff there is a path ~r0 such that:  1. rr0 is the &amp;quot;exception path&amp;quot; from the top node of the parallel structures the the two exception positions: v{=Vl.~ro A v~=v2.~ro 2. the two contexts, which are the trees below Vl and v2 up-to the trees below the exception positions v{ and v~, must have the same structure and labels:</Paragraph>
      <Paragraph position="9"> 3. there are no 'hanging' binders from the contexts to variables outside them:</Paragraph>
      <Paragraph position="11"> 4. binding is structurally isomorphic within the two contexts:  V rr V rr' -~ir o &lt; ~r A vl . Tr.L L A -~'tr o &lt;_Tr' A vl . lr' J~ L :=~ 5. two variables in identical positions within their context and bound outside their con- ~_.~.:y,. &amp;quot; text must be bound by the same binder: ,--~'~. I~-~ v,,w-(,,o&gt;,, /-'% :;*-1</Paragraph>
      <Paragraph position="13"> 6. two anaphors in identical positions within ~x ..</Paragraph>
      <Paragraph position="14"> their context must have isomorphic links x &amp;quot;.</Paragraph>
      <Paragraph position="15"> resents the semantics of the elided part of the target clause.)</Paragraph>
      <Paragraph position="17"> within their context, or the target sentence anaphor is linked to the source sentence anaphor:</Paragraph>
      <Paragraph position="19"/>
    </Section>
  </Section>
  <Section position="4" start_page="356" end_page="357" type="metho">
    <SectionTitle>
3 Interaction of quantifiers,
</SectionTitle>
    <Paragraph position="0"> anaphora, and ellipsis In this section, we will illustrate our analysis of a complex case of the interaction of scope, anaphora, and ellipsis. In the case (8), both anaphora and quantification interact with ellipsis. null  (8) Mary read a book she liked before Sue did. (8) has three readings (see (Crouch, 1995) for  a discussion of a similar example). In the first, the indefinite NP a book she liked takes wide scope over both clauses (a particular book liked by Mary is read by both Mary and Sue). In the two others, the operator before outscopes the indefinite NP. The two options result from the two possibilities of reconstructing the pronoun she in the ellipsis interpretation, viz., 'strict' (both read some book that Mary liked) and 'sloppy' (each read some book she liked herself).</Paragraph>
    <Paragraph position="1"> The constraint for (8), displayed in (18), is an underspecified representation of the above three readings. It can be derived in a compositional fashion along the lines described in (Niehren et al., 1997b). Xs and Xt represent the semantics of the source and the target clause, while X16 and X21 stand for the semantics of the parallel elements (Mary and Sue) respectively. For readability, we represent the semantics of the complex NP a book she liked by a triangle dominated by X2, which only makes the anaphoric content 212 of the pronoun she within the NP explicit. The anaphoric relationship between the pronoun she and Mary is represented by the linking relation between X12 and X16. (X20 rep-C/ read ~~7~1 ~Xz6 Xs/XI6~X~/X21 The first reading, with the NP taking wide scope, results when the relative scope between XI and XI5 is resolved such that XI dominates X15. The corresponding solution of the constraint is visualized in (19).</Paragraph>
    <Paragraph position="2"> (19) za, x=, read ~'~ var~-.X&amp;quot;z~ read ~ var~'~ j The parallelism constraint Xs/Xl6,,~Xt/X21 is satisfied in the solution because the node Xt dominates a tree that is a copy of the tree dominated by Xs. In particular, it contains a node labelled by var, which has to be parallel to Xlr, and therefore must be A-linked to X3 too.</Paragraph>
    <Paragraph position="3"> The other possible scoping is for XlS to dominate X1. The two solutions this gives rise to are drawn in (20) and (21). Here X1 and the interpretation of the indefinite NP directly below enter into the parallelism as a whole, as these nodes lie below the source node Xs. Thus, there are two anaphoric nodes: X12 in the source and its 'copy' II12 in the target semantics. For the copy to be parallel to XI2 it can either have a link to X12 to have a same referential value (strict reading, see (20)) or a link to X21 that is structurally parallel to the link from X12 to X16, and hence leads to the node of the parallel element Sue (sloppy reading, see (21)).</Paragraph>
    <Paragraph position="4">  (20) ~x, I&amp;quot;&amp;quot; ~&amp;quot;r, ary.,, X~6&amp;quot;~. ' ~/sue * _X</Paragraph>
  </Section>
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