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<Paper uid="J03-2002">
  <Title>c(c) 2003 Association for Computational Linguistics Implementing the Binding and Accommodation Theory for Anaphora Resolution and Presupposition Projection</Title>
  <Section position="4" start_page="189" end_page="192" type="intro">
    <SectionTitle>
3. Representing Discourse and Presuppositions
</SectionTitle>
    <Paragraph position="0"> This section is concerned with proper representations for discourse and presupposition, with the interpretation of discourse (aiming to being able to perform inferences), and further provides tools necessary for discourse processing and presupposition resolution. I will start with some formalities and define the syntax of standard DRSs and the syntax of a-DRSs (DRSs that contain unresolved presuppositions). I will argue that the representation for sentence-DRSs, as originally introduced for presuppositions in BAT, is insufficient for several interpretation tasks and introduce a new format.</Paragraph>
    <Section position="1" start_page="189" end_page="191" type="sub_section">
      <SectionTitle>
3.1 Representing and Interpreting Discourse
</SectionTitle>
      <Paragraph position="0"> DRSs capture the semantic content of a discourse. They form the medium for discourse understanding, because they come with a model-theoretical interpretation. The interpretation given here is one via a translation to first-order logic. This is advantageous from a practical and computational perspective, because one can use automated theorem provers for first-order logic to implement some of the acceptability constraints imposed by BAT or indeed carry out other kinds of inferences not related to anaphora resolution and presupposition accommodation.</Paragraph>
      <Paragraph position="1"> The syntax of DRSs and DRS-conditions is defined by simultaneuous recursion, with respect to a set of first-order variables and a vocabulary describing the predicate symbols and their respective arities.</Paragraph>
      <Paragraph position="2">  is a basic DRS-condition.</Paragraph>
      <Paragraph position="3"> 4. Every basic DRS-condition is a DRS-condition.</Paragraph>
      <Paragraph position="4"> 5. If B is a DRS, then !B, B, and B are DRS-conditions.</Paragraph>
      <Paragraph position="5"> 6. If B  are DRS-conditions.</Paragraph>
      <Paragraph position="6"> 7. If x is a variable and B is a DRS, then x : B is a DRS-condition. Given a DRS B = &lt;D, C&gt; , D is called the domain of B, members of C are the conditions of B, and members of D are called B's discourse referents. Clause 1 of the definition defines DRSs in the standard way. The basic conditions (clauses 2-3) are defined just as in standard DRT. Clause 5 introduces negation and the modal operators, and clause 6 disjunction and implication. Clause 7 is nonstandard; it introduces a modal operator that explicitly associates variables ranging over possible worlds with DRSs. It is therefore related to constructs used in hybrid logics (Blackburn 2000). We will use it in our fragment of English to represent sentential complements.</Paragraph>
      <Paragraph position="7"> DRSs are interpreted in an indirect manner, with the help of a translation function that maps DRSs to first-order formulas (under the same vocabulary of predicate symbols and with respect to the same set of variables). This translation is implemented as the function (.,.) fo , from first-order variables (ranging over possible worlds) and DRSs to ordinary first-order formula syntax. The complete translation is shown in the following definition.</Paragraph>
      <Paragraph position="9"> Bos Implementing Binding and Accommodation Theory This translation from DRSs to first-order logic is based on the one given in Kamp and Reyle (1993) extended with Moore's proposal for modal operators (Moore 1980). It behaves linearly on the size of the input, so the computational overhead is kept low. I will use it to implement the acceptability constraints imposed by BAT on presupposition resolution that require inference, to wit, the check for consistency and informativeness.</Paragraph>
    </Section>
    <Section position="2" start_page="191" end_page="192" type="sub_section">
      <SectionTitle>
3.2 Representing Presuppositions
</SectionTitle>
      <Paragraph position="0"> What is a proper representation for elementary presuppositions? There seems to be common agreement, in most of the accounts in presupposition theory, that presuppositions represent expressions of propositional type. Hence, to use a DRS to represent a single presupposition seems a natural choice.</Paragraph>
      <Paragraph position="1"> But there are further issues that play a role in deciding a suitable representation for presuppositions. Two operations on DRSs used in BAT are merge reduction and presuppositional binding,  and both require a precise definition of free and bound variables. However, sentence-DRSs allow &amp;quot;ambiguous&amp;quot; bindings. Consider, for instance, the DRS in example (23) (again I will underline the relevant presupposition triggers in the following examples): (23) A man smiles. The woman smokes.</Paragraph>
      <Paragraph position="3"> This sentence-DRS contains an A-structure with a single DRS. It is unclear whether the occurrence of variable x in the condition SMOKE(x) is bound by the discourse referent x in the outermost DRS or by the discourse referent declared in the DRS within the A-structure. Following the definition of sentence-DRSs, the discourse referent x in the A-structure does not in fact bind the occurrences of x in the main DRS. Furthermore, given the fact that A-structures can host more than one DRS, situations with ambiguous bindings might appear easily.</Paragraph>
      <Paragraph position="4"> These unintuitive and ambiguous bindings are unpleasant and force one to reconsider representing unresolved anaphoric expressions in DRT. The representation that I prefer uses a new operator, a, combining two DRSs to form a new DRS. This disallows ambiguous bindings while keeping the same expressive power: (24) A man smiles. The woman smokes.</Paragraph>
      <Paragraph position="6"> all bound occurrences of the anaphoric discourse referents are replaced by the variable name of the antecedent referent. This operation is preferred to adding an equality condition between the antecedent and anaphoric referent to the DRS, for two reasons: It decreases the search space for finding antecedents during subsequent instances of presupposition resolution, and it makes the inference problems derived from these DRSs less difficult.</Paragraph>
      <Paragraph position="7">  Computational Linguistics Volume 29, Number 2 I believe that the representation in example (24) is more intuitive as well, because it is presupposition that comes first in an utterance. The a operator reflects this, because its left argument is the presuppositional part, and its right argument the assertive part. Like A-structures, the a operator allows recursion and therefore nested presuppositions. null I would like to address another representational issue here. In Van der Sandt's  (1992) original formulation of BAT, all discourse referents appearing in the domain of a DRS in the A-structure are anaphoric. This leads to unexpected behavior, as exemplified by the sentence-DRSs for the two sentences in example (25): (25) The boy with a gun fires.</Paragraph>
      <Paragraph position="9"> The sentences in (25) contain noun phrases with restricted relative clauses. In the first example the relative clause contains an indefinite noun phrase a gun, whereas in the almost identical second example, it contains a definite (hence presuppositional) noun phrase. These two sentence-DRSs do not, however, reflect the difference in meaning of the utterances they represent. The indefinite noun phrase a gun gets an anaphoric interpretation, because it is part of the A-structure. The anaphoric potential of the two utterances, according to the sentence-DRSs for the two utterances in example (25), is almost identical, the only difference being that the DRS for the second utterance allows binding or accommodation on two different levels (instead of one level) of discourse structure. Summing up, A-structures do not allow for selective binding, with the unwanted side effect that indefinite noun phrases are turned into definite ones.</Paragraph>
      <Paragraph position="10"> To allow for selective binding, I introduce the notion of principal anaphoric referent. The operator a is indexed with the principal anaphoric referent to indicate which discourse referent of a presuppositional DRS is anaphoric. In fact, I assume that each presupposition trigger has a unique principal anaphoric referent.</Paragraph>
      <Paragraph position="11">  Now let us consider the DRSs for the same utterances in (25) in the new formats:</Paragraph>
    </Section>
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