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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="5" start_page="192" end_page="197" type="metho">
    <SectionTitle>
3 Although this seems a small and innocent adjustment to the representation of elementary
</SectionTitle>
    <Paragraph position="0"> presuppositions, it has great impact on our understanding what characterizes presuppositions. Under this view, presuppositions are not merely DRSs, but they are DRSs plus an additional pointer, in the form of a distinguished discourse referent, to a context.</Paragraph>
    <Paragraph position="1">  Bos Implementing Binding and Accommodation Theory (26) The boy with a gun fires. The boy with the gun fires.</Paragraph>
    <Paragraph position="3"> Although the entire DRS of the left-hand side of the a-operator is said to be presuppositional, resolution will affect only the principal discourse referent. For The boy with a gun, this is x, and only x is identified with an antecedent discourse referent; y, introduced for the indefinite noun phrase a gun, is not treated as anaphoric. This is in contrast to the DRS for The boy with the gun, which contains nested a-DRSs. In short, a allows selective binding, whereas Van der Sandt's A-structures are based on unselective binding.</Paragraph>
    <Paragraph position="4"> I will refer to this new format for DRSs, encoding unresolved anaphoric DRSs, as a-DRSs. Like sentence-DRSs, a-DRSs are intermediate representations and have no interpretation. The best way to view them is as underspecified representations encoding the ambiguities of anaphoric expressions in a compact manner. The syntax of a-DRSs is defined as follows:  )isana-DRS.</Paragraph>
    <Paragraph position="5"> Note that in clause 1 of the definition, ordered sets are used rather then plain sets for discourse referents and DRS-conditions; this will make definition of the resolution algorithm easier. DRS merging (clause 4) is used in many alternative formulations of DRT (Muskens 1996; Van Eijck and Kamp 1997; Kuschert 1999). The merge employed here, &amp;quot;;&amp;quot;, is adopted from Muskens and behaves semantically the same as dynamic conjunction in dynamic predicate logic (Groenendijk and Stokhof 1991).</Paragraph>
    <Paragraph position="6"> The syntax of a-DRS-conditions is defined as follows: Definition The syntax of a-DRS-conditions is defined according to the following four clauses:  1. Every basic DRS-condition is an a-DRS-condition.</Paragraph>
    <Paragraph position="7"> 2. If B is an a-DRS, then !B, B, and B are a-DRS-conditions.</Paragraph>
    <Paragraph position="8"> 3. If B  are a-DRS-conditions.</Paragraph>
    <Paragraph position="9"> 4. If x is a variable and B is an a-DRS, then x : B is an a-DRS-condition.  Computational Linguistics Volume 29, Number 2 So the syntax for a-DRSs subsumes the syntax of DRSs. I will refer to a-DRSs that contain no DRSs of the form (B a</Paragraph>
    <Paragraph position="11"> )aspresupposition-free. All other a-DRSs are referred to as presupposition-containing. (The same terminology will be used for a-DRS-conditions.) It is easy to show that presupposition-free a-DRSs are proper DRSs. In Table 1, I give several examples of lexical entries of presupposition triggers, assuming a compositional semantics in the Montagovian tradition based on Muskens's (1996) compositional DRT (Muskens, 1996). The syntactic categories used in the table are N (noun), DET (determiner), NP (noun phrase), ADJ (adjective), and V (verb).</Paragraph>
    <Paragraph position="12"> Further, p and q are used to denote variables ranging over properties, and s to denote variables ranging over propositions. The table lists various kinds of presupposition triggers, including the definite determiner, a proper name (Mia), a factive verb (to realize), a sortally restricted predicate (bachelor), and the iterative adjective other.</Paragraph>
    <Section position="1" start_page="194" end_page="197" type="sub_section">
      <SectionTitle>
3.3 Operations on Discourse Representations
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      <Paragraph position="0"> In this section I will formulate two operations on discourse representations required for presupposition resolution: renaming and merging. Furthermore, I will introduce the concepts of free and bound variables in discourse representations, because we need to implement the free-variable constraint. This will also give us a better understanding of the problems that arise in renaming and merging.</Paragraph>
      <Paragraph position="1"> A free variable in this context is an occurrence of a variable in a DRS that is not declared as a discourse referent in the domain of the immediate DRS in which it occurs or in the domain of a superordinated DRS; all other variables are bound.To be more precise, I define two functions, FREE and BOUND, that compute the free and bound variables for DRSs and DRS-conditions:</Paragraph>
      <Paragraph position="3"> Computational Linguistics Volume 29, Number 2 For renaming variables in DRSs, a (total) function r is used whose arguments are provided by an auxiliary (total) function s that maps a variable to a stringed sequence of a copy of itself. The use of s makes the values r assigns to a variable x sensitive to the number of occurrences of x in a DRS: s(x)=x if x has not appeared (bound) so far in a DRS, s(x)=xx if x has appeared (bound) once, xxx if x has appeared (bound) twice, and so on. Putting it differently, s is a counter. It counts the number of occurrences of declarations of discourse referents in a DRS. On the basis of that count, the renaming function r maps a variable onto a new, previously unused occurrence.</Paragraph>
      <Paragraph position="4"> The renaming function r maps a sequence of variables to a new, fresh discourse variable. The function r is injective, meaning that distinct argument sequences to r produce distinct values. In other words, the values r assigns to x, y, z, xx, yy, zz, xxx, yyy, zzz, and so on are all different. In sum, with the help of r and s, we make explicit that two discourse referents declared under the same name in two different domains in a DRS get two different names. The definition of renaming in DRS (where for a set of discourse referents U, UPDATE(s, U, s  The function RENAME maps an ordered pair consisting of a s-function and a DRS to a new s-function and the translated DRS. Clauses 1-3 of the definition handle DRSs, clauses 4-8 handle the DRS-conditions. With respect to variables not declared in a universe of a DRS, s remains unchanged. For variables that are declared as discourse referents in a DRS, s increases the values for these variables by one. Clauses 2, 3, and 6 show that the s-function produced by the first DRS is passed through to the second DRS, following the definition of DRS-subordination.</Paragraph>
      <Paragraph position="5"> With the renaming function at our disposal, we now can define pure DRSs as DRSs that have undergone renaming. The conversion of a DRSs potentially changes its discourse meaning, and this is actually the key function of renaming: maximizing the  Bos Implementing Binding and Accommodation Theory context change potential of a DRS. To preserve the logical meaning while renaming, we need to put restrictions on the use of renaming. As DRSs can bind variables outside their scope (for instance, through use of the merge operator), applying the renaming procedure only to a DRS B that is actually part of another DRS B prime would possibly affect the logical meaning of a DRS. Therefore, only complete DRSs should be renamed; that is, if a sub-DRSs is renamed, any DRS that superordinates it must be renamed too. Let me now relate this to practical discourse processing. Assume that processing a text proceeds in an incremental way, starting with processing the first sentence, until a DRS is eventually derived for the entire text. At some point during this process, after completing the analysis of a sentence, part of the text is translated into a DRS. At this stage of processing, the obtained DRS is complete (it is not part of another DRS, as the rest of the text is still unanalyzed), and it can be renamed without changing its logical meaning, while maximizing its context change potential. The rest of the discourse is then processed with respect to the renamed DRS.</Paragraph>
      <Paragraph position="6"> Hence, given a pure DRS, we can replace a DRS (B; B prime ) with a new DRS that is constructed by taking the unions of the discourse referents and the conditions of B and B prime , respectively. We will specify the merging of DRSs with the help of a function MERGE from DRSs to DRSs. This function is defined recursively:  Carrying out merge reduction simplifies the DRS structure and facilitates use of the standard accessibility definition. Moreover, using MERGE, it is straightforward to define the set of discourse referents within a universe of a pure DRS B, namely, U</Paragraph>
      <Paragraph position="8"> I will make use of this when I implement pronoun and presupposition resolution in the next section. Finally, merge reduction yields DRSs that can be transformed into first-order logic using the translation function given in Section 3.1.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="197" end_page="204" type="metho">
    <SectionTitle>
4. Presupposition Resolution
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    <Paragraph position="0"> In this section I will reformulate Van der Sandt's presupposition resolution algorithm in terms of a-DRSs with the aim of reducing the generate-and-test nature of Van der Sandt's original formulation. Even for relative simple examples, the search space of possible interpretations is vast. Consider the following example: (27) If Bonnie finds a corpse in her house, the dead body will frighten her.</Paragraph>
    <Paragraph position="1"> Without applying any acceptability constraints, resolving the four triggers in example (27) will yield 1,280 different solutions. If binding is preferred to accommodation, the example will still give rise to 525 possible interpretations. It is clear that a generate-and-test approach, in which the acceptability constraints are applied to completely resolved DRSs, will be extremely inefficient. The algorithm implemented in this section applies, as far as possible, the acceptability constraints to partially resolved DRSs, and thereby reduces the search space.</Paragraph>
    <Paragraph position="2">  Computational Linguistics Volume 29, Number 2 To deal with the different anaphoric behavior of noun phrases in English, I will propose a typology of noun phrases reflecting their properties with respect to binding and accommodation. Further, I will precisely formulate the acceptability constraints imposed by the resolution algorithm and outline how to add preference ranking to the solutions generated by the algorithm.</Paragraph>
    <Section position="1" start_page="198" end_page="200" type="sub_section">
      <SectionTitle>
4.1 The Resolution Algorithm
</SectionTitle>
      <Paragraph position="0"> Presupposition resolution in discourse processing is assumed to take place on an utterance-by-utterance basis. Therefore, the resolution algorithm takes as input the DRS constructed for the discourse so far and an a-DRS for the new utterance and outputs a new DRS. The algorithm is recursive in nature, and given an a-DRS A with n presupposition triggers, each step in the resolution process will resolve one trigger and decrease n by one, until A is presupposition-free (i.e., n = 0).</Paragraph>
      <Paragraph position="1"> The idea behind the resolution algorithm is to detect any violations of the acceptability constraints as soon as possible in the process of resolution, thereby restricting the search space. After each resolution step the acceptability constraints are checked for violation. Some of the acceptability constraints, however, are not defined for a-DRSs and can be applied only to the finally resolved DRS.</Paragraph>
      <Paragraph position="2"> I will present the algorithm in a notation borrowed from logic programming, using negation as failure, backtracking for efficiency, and unification for term manipulation.</Paragraph>
      <Paragraph position="3"> Given the definition below, it is straightforward to implement the algorithm using a programming language like PROLOG. DRSs are represented as an ordered pair of lists, and PROLOG-style variables are used to represent discourse referents and first-order variables. Further, I will use the following notational conventions: A, B, and C are used to denote DRSs, L and M to denote DRS-conditions, and P and Q to denote accessibility paths.</Paragraph>
      <Paragraph position="4"> The main predicate of the algorithm consists of the following two clauses: resolve(C,B,B) - presup-free(B), consistent(B), informative(C,B).</Paragraph>
      <Paragraph position="5"> resolve(C,B,E) - alfa(B-D,[]-[A|P]), move(A,P,D), resolve(C,D,E).</Paragraph>
      <Paragraph position="6"> The first resolve clause terminates the recursion if all presuppositions are resolved, then checks the resulting DRS for consistency and informativeness (see Section 4.3). The second, recursive resolve clause makes use of two further predicates that I introduce now: alfa, for determining the first presupposition trigger in the a-DRS, and move, which either binds or accommodates the trigger to an accessible level of discourse.</Paragraph>
      <Paragraph position="7"> Let us consider first alfa, which is defined for a pair of a-DRSs (or a-DRSconditions) and a pair of accessibility paths. (Recall that an accessibility path is a list of levels of DRSs, starting with the presupposition trigger and ending with the global level of discourse.) This is the definition for a-DRSs:</Paragraph>
      <Paragraph position="9"> What alfa effectively does is traverse the DRS structure, thereby building up the accessibility path, until it hits a presupposition trigger. The accessibility path is constructed as a list of binding or accommodation sites. Binding sites are marked as bin(A,B), where A is the original site (i.e., a DRS) and B the result of binding. Accommodation sites  Bos Implementing Binding and Accommodation Theory are marked as acc(A), where A is the result of accommodation. For instance, consider the two clauses for the DRS merge in the definition above, in which the input DRS is of the form (A;B). If A contains a presupposition, then alfa(A-C,P-Q) holds, and the resulting DRS will be set to (C;B). If A is presupposition-free, B will be traversed for presuppositions (resulting in C) and a binding site X for A will be introduced on the accessibility path represented by P and Q. Possible binding sites are also introduced by basic DRSs. Basic DRSs furthermore introduce a possible accommodation site in case one of their complex conditions contain a-DRSs. The clauses for DRS-conditions are defined as follows:</Paragraph>
      <Paragraph position="11"> alfa([A=B|L]-[(E;D)=C|L],P-Q)- presup-free(A), presup-free(A), alfa(BC,[bin(A,E),acc(D)|P]-Q). null Note that, because we use ordered sets of DRS-conditions, the predicate alfa behaves in a deterministic way, and it returns the first presuppositional DRS that itself is presupposition-free. Further note that the accessibility path reflects the accessibility relation defined in DRT, which is mirrored by the clauses for alfa. For instance, note the difference in definition between the implicational and disjunctional condition.</Paragraph>
      <Paragraph position="12"> The accessibility path returned by alfa forms a skeleton for a resolved DRS, which will be instantiated based on the decision as to whether to bind or accommodate the presupposition, and on which level. For a site encoded by acc(A), accommodation involves identifying the presuppositional DRS with A. For binding, it involves matching the presupposition with DRS A resulting in a new DRS B for a site of the form bin(A,B). This process is implemented by the predicates move, bind, accommodate, and skip. Let us first consider move. As BAT dictates, resolution involves either binding or accommodation: move(A,P,B) - bind(A,P), !sortal-violation(B), !binding-violation(B).</Paragraph>
      <Paragraph position="13"> move(A,P,B) - accommodate(A,P), !free-variables(B).</Paragraph>
      <Paragraph position="14"> The first clause of move binds the presupposition to a DRS on the accessibility path and then checks the acceptability constraints (see Section 4.3 for the definition of these constraints). The second clause invokes accommodation, followed by a check on free variables (again, see Section 4.3). Let us first consider binding. Binding is possible only on binding sites, where the presuppositional information is matched with the DRS on the binding site, resulting in a new DRS. Binding is defined recursively, for there might be several appropriate binding sites:  Computational Linguistics Volume 29, Number 2 down to unification of the presuppositional information with the DRS on the accommodation site. Again, there might be several accommodation sites (corresponding to global, local, or intermediate levels of discourse), so a recursive definition is appropriate: null accommodate(A,[acc(A)|P]) - skip(P).</Paragraph>
      <Paragraph position="15"> accommodate(A,[S|P]) - accommodate(A,P), skip([S]).</Paragraph>
      <Paragraph position="16"> Finally, we need to define skip. This function takes care of all accommodation and binding sites that were not selected as antecedent for the presupposition trigger. For a possible accommodation site, this will result in identifying its DRS with an empty DRS, and for a binding site, the resulting DRS will be unchanged with respect to its original one: skip([]) - true.</Paragraph>
      <Paragraph position="17"> skip([acc(&lt;[?],[?]&gt; )|P]) - skip(P).</Paragraph>
      <Paragraph position="18"> skip([bin(A,A)|P]) - skip(P).</Paragraph>
      <Paragraph position="19"> This is the core of the algorithm, but various extensions are possible. In the following section I will take different types of presuppositional triggers into account, because some expressions allow for accommodation and others do not, and some can be resolved only at the global level of discourse, whereas others are not sensitive to subordinated levels. Furthermore, I will formulate the acceptability constraints and investigate means to account for preferences in solutions. (As it stands, the algorithm will produce a set of solutions, all equal to one another. But as noted in the discussion of BAT, there are sometimes clear preferences for certain solutions.)</Paragraph>
    </Section>
    <Section position="2" start_page="200" end_page="201" type="sub_section">
      <SectionTitle>
4.2 Classifying Presupposition Triggers
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      <Paragraph position="0"> The resolution algorithm, as formulated in the previous section, does not discriminate among different types of anaphoric expressions. With regard to noun phrases, it is well known that the choice of referring expression affects coherence in discourse (Grosz, Joshi, and Weinstein 1995). Perhaps related to this observation is the fact that pronouns, definite descriptions, and proper names all vary in terms of their capacity for binding and accommodation. The performance of the algorithm would strongly benefit from taking these differences into account, because it would further narrow down the search space.</Paragraph>
      <Paragraph position="1"> With respect to accommodation, some noun phrases allow accommodation on any level, whereas others accommodate only globally. Third-person anaphoric pronouns normally do not allow accommodation, with the exception of discourse-initial occurrences. Reflexive pronouns do not have the ability to accommodate, for they are intrinsically anaphoric. Definite descriptions, especially genitive constructions, have the power to accommodate on all levels (see example (7)). Proper names allow accommodation only on the global level. If one also considers first- and second-person pronouns, which belong to the family of deictic expressions, it can be concluded that this class of expressions does not allow accommodation at all, simply because deictic expressions refer to objects presumed in the context of interpretation.</Paragraph>
      <Paragraph position="2"> For binding, the differences among noun phrases are not so marked. Most of them allow binding on all levels of discourse structure, with the exception of proper names.</Paragraph>
    </Section>
    <Section position="3" start_page="201" end_page="202" type="sub_section">
      <SectionTitle>
Local Global
a-type Binding/Accommodation Binding/Accommodation Description
</SectionTitle>
      <Paragraph position="0"> ref yes/no yes/no reflexive pronouns pro yes/no yes/yes third-person nonreflexive pronouns nam no/no yes/yes proper names dei no/no yes/no first- and second-person nonreflexive pronouns def yes/yes yes/yes definite descriptions Antecedents of deictic expressions are assumed to be available at the global level of discourse, for they are part of the current context of interpretation, and so reference to objects at subordinated levels of discourse is not an option for deictic expressions. To account for the different referential behavior of noun phrases, we classify them in terms of a-types. The a-types for English noun phrases and their properties are listed in Table 2. To integrate these properties into the resolution algorithm, we need a way to determine whether we resolve a particular presupposition at a global or nonglobal level of discourse. Given an accessibility path, it is unequivocal to define the conditions for operating on a global level of discourse, because subordinated levels of DRS introduce an accommodation site, which we represent as acc(A) for a DRS A, on the accessibility path. Therefore, a binding site is global if there is no accommodation site on the remainder of its accessibility path: global(P) -!acc(A)[?]P.</Paragraph>
      <Paragraph position="1"> With this machinery we are able to revise the definition for accommodation, by making it sensitive to different a-types. This results in the following clauses:</Paragraph>
      <Paragraph position="3"> Similarly, we can revise the definition of binding by making it sensitive to different a-types. This yields the following clauses:  This might be a rather rocky approximation to discriminating among different noun phrases, but it will greatly improve the performance of the algorithm. Whether a finer classification is required, or whether further types to deal with other kinds of presupposition triggers (such as factives) are needed, remains subject for future corpus studies.</Paragraph>
      <Paragraph position="4">  Computational Linguistics Volume 29, Number 2</Paragraph>
    </Section>
    <Section position="4" start_page="202" end_page="202" type="sub_section">
      <SectionTitle>
4.3 Acceptability Constraints
</SectionTitle>
      <Paragraph position="0"> The resolution algorithm imposes several acceptability constraints on resolved or partially resolved DRSs. For completely resolved DRSs, there are constraints on consistency and informativeness. For partially resolved DRSs (i.e., a-DRSs), there are constraints on sortal compatibility, binding, and the occurrences of free variables. Let us first consider consistency and informativeness.</Paragraph>
      <Paragraph position="1"> As an illustration of the constraints on consistency, suppose we have a DRS B.</Paragraph>
      <Paragraph position="2"> If we can prove that ![?]w(w,B) fo is valid, then we know that B is inconsistent. If, on the other hand, we find that [?]w(w,B) fo is satisfiable, we know that B is consistent. In terms of our previous formulation of the resolution algorithm, this translates as  The constraint on sortal compatibility can be seen as a local consistency check. It takes place after binding, and it uses a sortal ontology to ensure that discourse referents with different sorts are not identified with each other. This eliminates any possibility for anaphoric expressions that describe entities to refer to discourse referents for temporal information or possible worlds and so cuts the search space of antecedents enormously. In terms of the algorithm, sortal-violation(B) will hold for a DRS B if there is an accessibility path in B with a discourse referent that has inconsistent properties.</Paragraph>
      <Paragraph position="3"> The binding constraint is a linguistic confinement primarily dealing with restrictions of antecedents of anaphoric object noun phrases. Binding constraints are similar to the C-command constraints found in linguistic theory, but I will give a simplified formulation here and deal with (di)transitive verbs only. Binding constraints deal with two complementary cases. First of all, it checks whether a reflexive pronoun in object position is bound to the subject noun phrase. Second, it checks whether a nonreflexive anaphoric noun phrase in object position is not bound to the subject noun phrase. Finally, there is the constraint on bound variables for a-DRSs. With the help of the definition of free and bound variables given in Section 3.3, it is straightforward to include this constraint in the resolution algorithm. This can be accomplished efficiently by traversing the DRS in a top-down manner, collecting bound variables on each accessibility path. Free variables are then detected when one of the variables occurring in a basic condition is not a member of the set of bound variables collected on that accessibility path.</Paragraph>
    </Section>
    <Section position="5" start_page="202" end_page="204" type="sub_section">
      <SectionTitle>
4.4 Preferences
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      <Paragraph position="0"> In this section I will investigate how to account for ranking interpretations. The resolution algorithm will produce a set of solutions without stating any preferences among  Bos Implementing Binding and Accommodation Theory candidates in the solution set produced. In general, it has been noted that in most cases, binding is strongly preferred to accommodation (Van der Sandt 1992), and that global accommodation is preferred to local accommodation (Heim 1983). This meshes well with a claim put forward in theories of discourse coherency that the inference demands placed on a hearer correlate positively with the perceived coherency of a discourse (Grosz, Joshi, and Weinstein 1995), because it makes perfect sense to ascribe a higher inference load to accommodation than binding.</Paragraph>
      <Paragraph position="1"> Given the resolution algorithm as defined here, one way to invoke a ranking mechanism among potential solutions is to include scores and make them sensitive to different a-types. Scores could be represented as numbers between 0 and 1, reflecting the rank of the solution in the solution set. Starting with a score of 1 for a particular solution, accommodation will decrease the score (for instance, by multiplying the current score by 0.1), whereas binding will not. Cases of nonglobal accommodation will further lower the score.</Paragraph>
      <Paragraph position="2"> Van der Sandt's constraints on local informativeness and local consistency are further criteria for preference ranking. Unlike the global versions of informativeness and consistency, the local constraints cannot be &amp;quot;hard&amp;quot; constraints, for if they were, they would rule out otherwise fine solutions. Put differently, rejection of DRSs on the basis of violating the local informativeness constraint seems inappropriate.</Paragraph>
      <Paragraph position="3"> I will illustrate this observation with the discourse and its translation in a DRS (example (28)).</Paragraph>
      <Paragraph position="4">  The DRS in example (28) violates the local informativeness constraint. The sub-DRS containing the information that there is a woman is already expressed by the superordinated DRS, given the background knowledge that Mia is a woman. From a grammatical point of view, however, example (28) is a legitimate sentence. Rejecting it on the basis of violating local informativeness seems unjustified. On the other hand, the local constraints help in dealing with the presupposition project problem. Therefore, we take a suggestion made by Beaver (2002) and use the local constraints as a further criterion for ranking potential solutions of the resolution algorithm. This ranking could be realized by decreasing the score of a particular DRS each time it violates local informativeness or local consistency.</Paragraph>
      <Paragraph position="5"> Incidentally, Van der Sandt does not give a precise formulation of local informativeness and local consistency, and it is not straightforward what would constitute a precise formulation (Beaver 1997, 2002). I will give a novel formulation of the local constraints with the help of a function that, given a DRS, returns a set of pairs of DRSs and the DRSs that they subordinate. Given this function, it is straightforward to define local informativeness and local consistency. This function, supersub, is defined, using PROLOG notation, as follows (the definition given here is restricted to the clauses for implication, negation and basic conditions; clauses for the remaining conditions can  Computational Linguistics Volume 29, Number 2 be easily derived from these):</Paragraph>
      <Paragraph position="7"> The supersub predicate is recursively defined to handle arbitrarily deeply embedded sub-DRSs. The crucial step in supersub is removing the conditions that contain the subordinated DRS. Applying supersub to example (28), we get the following two pairs:  We can now formulate the local constraints in terms of informativeness as follows: A pair &lt;super-DRS: A, sub-DRS B&gt; is locally informative if (A;B) is informative with respect to A and locally consistent if (A;&lt;[?],[!B]&gt; ) is informative with respect to A. Adding the local constraints as further criteria for ranking potential readings produced by the resolution algorithm will give accurate predictions for the interpretation of many problematic cases discussed in Section 2.2. Testing the local constraints involves first-order theorem proving, and it is useful to add some heuristics to obtain more efficient implementations. A valuable heuristic is one that distinguishes subordinated levels of discourse in the old DRS (i.e., the DRS capturing the portion of the discourse processed so far) from subordinated levels of discourse in the DRS of the newly processed utterance. This avoids repeated application of local constraints to the same subordinated levels of discourse over and over again.</Paragraph>
    </Section>
  </Section>
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