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<Paper uid="P02-1010">
  <Title>Ellipsis Resolution with Underspecified Scope</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 Underspecified Discourse
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
Representation Structures
</SectionTitle>
      <Paragraph position="0"> Reyle (1993) proposes a formalism for under-specification of scope ambiguity. The under-specified representations are called Underspecified Discourse Representation Structures (UDRSs). Completely specified UDRSs correspond to the Discourse Representation Structures (DRSs) of Kamp and Reyle (1993). A UDRS is a triple consisting of a top label, a set of labelled conditions or discourse referents, and a set of subordination constraints. A UDRS is (partially) disambiguated by adding subordination constraints. A UDRS must, however, always comply with the following well-formedness conditions: (1) It does not contain cycles (subordination is a partial order). (2) No label is subordinated to two labels which are siblings, i.e.</Paragraph>
      <Paragraph position="1"> part of the same complex condition (subordination is a tree order).</Paragraph>
      <Paragraph position="2"> Figure 1 shows the UDRS for sentence 4 in formal  and graph representation.</Paragraph>
      <Paragraph position="3"> (4) Every professor found most solutions.</Paragraph>
      <Paragraph position="5"> type of constraint is introduced, accessibility constraints. a33a35a34 is accessible from a33a37a36 (a33a38a36 acc a33a35a34 ) iff a33a39a36a41a40 a33a14a34 or a42a43a33a35a44a46a45a47a33a38a36a48a40a49a33a14a44 and a33a35a44 is a right sibling of a33a50a34 in a condition expressing material implication or a generalized quantifier (Kamp and Reyle, 1993). An accessibility constraint a33a51a36 acc a33a35a34 indicates that a33a35a34 is an anaphoric element or a presupposition; it thus can be used as a trigger for anaphora resolution and presupposition binding (van der Sandt, 1992). To bind an anaphor a33a50a34 to some antecedent expression a33a38a44 , a binding constraint (a33 a34a53a52a54 a33 a44 ) and an equality constraint between two discourse referents are introduced. Binding constraints are interpreted as equality in the subordination order. Any unbound presuppositions remaining after anaphora resolution (corresponding to accessibility constraints without binding constraints) are accommodated, i.e. they end up in an accessible scope position which is as near to the top as possible. Figure 2 shows the UDRS for sentence (5). Accessibility constraints are marked by broken lines, binding constraints are shown as squiggles.</Paragraph>
      <Paragraph position="6">  (5) John revised his paper.</Paragraph>
      <Paragraph position="7"> l7: revise( x, y )</Paragraph>
      <Paragraph position="9"/>
    </Section>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Ellipsis Resolution
</SectionTitle>
    <Paragraph position="0"> Sag (1976) and Williams (1977) have argued convincingly that VP ellipsis should be resolved on a level where scope is fixed. Dalrymple et al. (1991) distinguish two tasks in ellipsis resolution: 1. determining parallelism, i.e. identifying the source clause a7 a8 (the antecedent of the ellipsis), the parallel elements in the source clause</Paragraph>
    <Paragraph position="2"> 2. interpreting the elliptical (target) clause a7 a15 , given the interpretation of</Paragraph>
    <Paragraph position="4"> The paper does not have much to say about task 1.</Paragraph>
    <Paragraph position="5"> Rather, some &amp;quot;parallelism&amp;quot; module is assumed to take care of task 1. This module determines the UDRS representations of the source clause and of the source and target parallel elements. It also provides a bijective function a23 associating the parallel labels and discourse referents in source and target.</Paragraph>
    <Paragraph position="6"> For task 2 we adopt the substitutional approach advocated by Crouch (1995): The semantic representation of the target a7 a15 is a copy of the source a7 a8 where target parallel elements have been substituted for source parallel elements (a7 a15a25a24</Paragraph>
    <Paragraph position="8"> Unification (HOU) (Dalrymple et al., 1991) substitution is deterministic: Ambiguities somehow cropping up in the interpretation process (i.e. the strict/sloppy distinction) require a separate explanation. null</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 Scope Parallelism
</SectionTitle>
    <Paragraph position="0"> It has frequently been observed that structural ambiguity does not multiply in contexts involving ellipsis: A scope ambiguity associated with the source must be resolved in the same way in source and target. Sentence (6) e.g. has no reading where all professors found the same solution but the students who found a solution each found a different one.</Paragraph>
    <Paragraph position="1"> (6) Every professor found a solution, and most students did, too.</Paragraph>
    <Paragraph position="2"> Scope parallelism seems to be somewhat at odds with the idea of resolving ellipses on scopally under-specified representations. If the decisions on scope order have not yet been taken, how can they be guaranteed to be the same in source and target? The QLF approach (Crouch, 1995) gives an interesting answer to this question: It uses re-entrancy to propagate scope decisions among parallel structures.</Paragraph>
    <Paragraph position="3"> In sentence (6), we see that a scope decision can resolve more than one ambiguity. In UDRT, scope decisions are modelled as subordination constraints.</Paragraph>
    <Paragraph position="4"> Consequently, sentence (6) shows that subordination constraints may affect more than one pair of labels. Remember that in each process of ellipsis resolution a32 the parallelism module returns a bijective function a23 a2 which expresses the parallelism between labels and discourse referents in source and target. As sentence (6) shows, a subordination constraint that links two source labels a33 a36 and a33 a34 also links the labels corresponding to a33a51a36 and a33a14a34 in a parallel structure a32 , i.e. a23 a2a34a33 a33a38a36a28a35 and a23 a2a36a33 a33a35a34a17a35 for all a32 . Thus the subordination constraint does not distinguish between source label and parallel labels. Formally, we define two labels a33 a36 and a33 a34 to be equivalent (a33 a36a38a37 a33 a34 )</Paragraph>
    <Paragraph position="6"> allelism effects by stipulating that a subordination constraint connects two equivalence classes a26a33 a36 a30a48a47 and a26a33a14a34 a30a48a47 rather than two individual labels a33 a36 and a33a35a34 . But every label in one class should not be linked to every label in the other class. If a33a51a36 and a33a14a34 are the source labels, it does not make sense, and actually will often lead to a structure violating the well-formedness conditions, to connect e.g. the source label a33 a36 with some target label a23 a2 a33 a33 a34 a35 . Thus we still need a proviso that only such labels can be linked that were determined to be parallel to the source label in the same sequence of ellipsis resolutions. We talk about a sequence here, because, as sentence (7) shows, ellipses may be nested.</Paragraph>
    <Paragraph position="7"> (7) John arrived before the teacher did (1 arrive), and Bill did too (2 arrive before the teacher did (1 arrive)).</Paragraph>
    <Paragraph position="8"> For the implementation of classes, we take our cues from Prolog (Erbach, 1995; Mellish, 1988). In Prolog, class membership is most efficiently tested via unification. For unification to work, the class members must be represented as instances of the representation of the class. If class members are mutually exclusive, their representations must have different constants at some argument position. In this vein, we can think of a label as a Prolog term whose functor denotes the equivalence class and whose argument describes the sequence of ellipsis resolutions that generated the label. Such a sequence can be modelled as a list of numbers which denote resolutions of particular ellipses. An empty list indicates that the label was generated directly by semantic construction. We will call the list of resolution numbers associated with a label the label's context. For reasons that will become clear only in section 7 discourse referents also have contexts.</Paragraph>
    <Paragraph position="9"> Although subordination constraints connect classes of labels, they are always evaluated in a particular context. Thus a33a37a36 a40 a33a1a0 (or, more explicitly, a2a4a3 a45</Paragraph>
    <Paragraph position="11"> case context changes.</Paragraph>
    <Paragraph position="12"> While scope resolution is subject to parallelism and binding is too (see Section 7), examples like (9) suggest that accommodation sites need not be parallel1. (&amp;quot;The jeweler&amp;quot; is accommodated with wide 1Asher et al. (2001) use parallelism between subordination and accommodation to explain the &amp;quot;wide-scope puzzle&amp;quot; observed by Sag (1976). Sentence (8) has only one reading: A specific nurse saw all patients.</Paragraph>
    <Paragraph position="13">  (8) A nurse saw every patient. Dr. Smith did too.</Paragraph>
    <Paragraph position="14"> scope, but &amp;quot;his wife&amp;quot; is not.) (9) If Peter is married, his wife is lucky and the jeweler is too.</Paragraph>
    <Paragraph position="15">  Ellipsis resolution works as follows. In semantic construction, all occurrences of labels and discourse referents (except those in subordination constraints) are assigned the empty context (a26a30 ). Whenever an occurrence of ellipsis is recognized, a counter is incremented. Let a32 be the counter's new value. All parallel labels a33 and discourse referents a7 in the target are replaced by their counterparts in the source</Paragraph>
    <Paragraph position="17"> is added to the context of every label and discourse referent in a7 a15 . Finally, the non-parallel target ele-</Paragraph>
    <Paragraph position="19"> tic representation of the target. Figure 3 shows the UDRS for sentence (6) after ellipsis resolution.</Paragraph>
    <Paragraph position="21"> Erk and Koller (2001) discuss sentence (10) which has a reading in which each student went to the station on a different bike. The example is problematic for all approaches which assume source and target scope order to be identical (HOU, QLF, CLLS).</Paragraph>
    <Paragraph position="22"> (10) John went to the station, and every student did too, on a bike.</Paragraph>
    <Paragraph position="23"> Erk and Koller (2001) go on to propose an extension of CLLS that permits the reading. In the approach proposed here no special adjustments are needed: The indefinite NP is designated by labels that do not have counterparts in the source. The subordination order is still the same in source and target.</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 Antecedent-Contained Ellipsis
</SectionTitle>
    <Paragraph position="0"> The elliptical clause can also be contained in the source, cf. example (11).</Paragraph>
    <Paragraph position="1"> (11) John greeted every person that Bill did. In this case the quantifier embedding the elliptical clause necessarily takes scope over the source. The treatment of this phenomenon in QLF and CLLS, which consists in checking for cyclic formulae after scope resolution, cannot be transferred to UDRT, since it presupposes resolved scope. Rather we make a distinction between proposed source and actual source. If the target is not contained in the (proposed) source, the actual source is the proposed source. Otherwise, the actual source is defined to be that part of the proposed source which is potentially subordinated2 by the nuclear scope of the quantifier whose restriction contains the target.</Paragraph>
  </Section>
  <Section position="8" start_page="0" end_page="121" type="metho">
    <SectionTitle>
6 Interaction of Ellipsis Resolution and
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="121" type="sub_section">
      <SectionTitle>
Quantifier Scoping
</SectionTitle>
      <Paragraph position="0"> Sentence (6) has a third reading in which the indefinite NP &amp;quot;a solution&amp;quot; is raised out of the source clause and gets wide scope over the conjunction. In this case, the quantifier itself is not copied, only the bound variables which remain in the source. Generally, a quantifier that may or may not be raised out of the source is only copied if it gets scope inside the source. Thus the exact determination of the semantic material to be copied (i.e. of the source) is dependent on scope decisions. Consequently, in an approach working on fully specified representations (Dalrymple et al., 1991) scope resolution cannot simply precede ellipsis resolution but rather is interleaved with it. Crouch (1995) considers ordersensitivity of interpretation a serious drawback. In his approach, underspecified formulae are copied in ellipsis resolution. In such formulae, quantifiers are not expressed directly but rather stored in &amp;quot;q-terms&amp;quot;. Q-terms are interpreted as bound variables. Quantifiers are introduced into interpreted structure only when their scope is resolved. Since scope resolution is seen as constraining the structure rather than as an operation of its own, the QLF approach manages to 2a1 a13 is potentially subordinated to a1a16a15 in a UDRS iff the subordination constraint a1 a13 a19a20a1 a15 could be added to the UDRS without violating well-formedness conditions.</Paragraph>
      <Paragraph position="1"> untangle scope resolution and ellipsis resolution. In CLLS (Egg et al., 2001) no copy is made in the underspecified representation. In both approaches, the quantifier is not copied until scope resolution.</Paragraph>
      <Paragraph position="2"> But the Missing Antecedents phenomenon (1) shows that a copy of the quantifier must be available even before scope resolution so that it can serve as antecedent. But this copy may evaporate later on when more is known about the scope configuration. We will call conditions that possibly evaporate phantom conditions. For their implementation we make use of the fact that a UDRS collects labelled conditions and subordination constraints in sets. In sets, identical elements collapse. Thus, a condition that is completely identical to another condition will vanish in a UDRS. Phantom conditions only arise by parallelism; hence they are identical to their originals but for the context of their labels and discourse referents. To capture the effect of possible evaporation, it suffices to make the update of context in a phantom condition dependent on the relevant scope decision. To implement phantom conditions in a Prolog-style environment, we insert a meta-variable in place of the context and control its instantiation by a special constraint expressing the dependence on the pertinent subordination constraint (a conditional constraint). Conditional constraints have the form a0 a24 a33 a33 a36 a40 a33 a34a2a1 a26a32a4a3 Ka30 a45 Ka35 where a0 is the context variable, a32 is a resolution number, and K is some context.</Paragraph>
      <Paragraph position="4"> representation of this UDRS can be seen in the first conjunct of Figure 5. Contexts are marked by dotted boxes, conditional constraints by a dotted subordination link with an equation.</Paragraph>
      <Paragraph position="5"> If the subsequent discourse contains a plural anaphoric NP such as &amp;quot;both solutions&amp;quot;, two or more discourse referents designating solutions are looked for. Two such discourse referents are found (a0 a26a30 and</Paragraph>
      <Paragraph position="7"> After consultation of the conditional constraint, the subordination constraint a33a2a1a47a40 a33 a0 is added. If the subsequent discourse contains a singular anaphoric NP &amp;quot;the solution&amp;quot;, anaphora resolution introduces the converse subordination constraint a33a2a3a47a40 a33 a1 .</Paragraph>
      <Paragraph position="8"> Examples involving nested ellipsis (cf. sentence (12)) require copying of context variables and conditional constraints.</Paragraph>
      <Paragraph position="9"> (12) Every professor found a solution before most students did, and every assistant did too.</Paragraph>
      <Paragraph position="10"> To copy a context variable a0 , it is replaced by a new variable a4 . The conditional constraint evaluating a0</Paragraph>
      <Paragraph position="12"> a35 ) is copied to a conditional constraint evaluating a4 . In this constraint a4 is conditionally bound back to a0 : a4</Paragraph>
      <Paragraph position="14"> a0 a35 , where a32 is the new resolution number and a33 a36 a36 is the top label of the source. Consider the UDRS for sentence (12) in Figure 5 with three conditional constraints: a0 a24 a33 a33 a1 a40 a33a1a0 a1 a26a6a5 a30 a45 a26a30 a35 , a5 a24 a33 a33 a1 a40 a33a39a36 a36 a1 a26a7a6 a30 a45</Paragraph>
      <Paragraph position="16"> istential NP &amp;quot;a solution&amp;quot; is copied three times (if a33 a1 a40 a33 a0 ), once (if a33 a1a9a8 a33a1a0 and a33 a1 a40 a33a39a36 a36 ), or not at all (if a33 a1a10a8 a33a39a36 a36 ).</Paragraph>
    </Section>
  </Section>
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