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<Paper uid="P01-1061">
  <Title>Computational properties of environment-based disambiguation</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 Representation of referents
</SectionTitle>
    <Paragraph position="0"> Existing environment-based methods (such as those proposed by Winograd) only calculate the referents of noun phrases, so they only consult the objects in an environment database when interpreting input sentences. But the evaluation of ambiguous sentences will be incomplete if the referents for verb phrases and other predicates are not calculated. In order to evaluate the possible interpretations of a sentence, as described in the previous section, an interface needs to define referent sets for every possible constituent.2 The proposed solution draws on a theory of constituent types from formal linguistic semantics, in which constituents such as nouns and verb phrases are represented as composeable functions that take entitiess or situations as inputs and ultimately return a truth value for the sentence. Following a straightforward adaptation of standard type theory, common nouns (functions from entities to truth values) define potential referent sets of simple environment entities: a9a11a10a13a12a15a14a16a10a15a17a18a14a16a10 a5 a14a20a19a20a19a20a19a22a21 , and sentences (functions from situations or world states to truth values) define potential referent sets of situations in which those sentences hold true:</Paragraph>
    <Paragraph position="2"> application, these situations can be represented as intervals along a time line (Allen and Ferguson, 1994), or as regions in a three-dimensional space (Xu and Badler, 2000), or as some combination of the two, so that they can be constrained by modifiers that specify the situations' times and locations. Referents for other types of phrases may be expressed as tuples of entities and situations: one for each argument of the corresponding logical function's input (with the presence or absence of the tuple representing the boolean output). For example, adjectives, prepositional phrases, and relative clauses, which are typically represented as situationally-dependent properties (functions from situations and entities 2This is not strictly true, as referent sets for constituents like determiners are difficult to define, and others (particularly those of quantifiers) will be extremely large until composed with modifiers and arguments. Fortunately, as long as there is a bound on the height in the tree to which the evaluation of referent sets can be deferred (e.g. after the first composition), the claimed polynomial complexity of referent annotation will not be lost.</Paragraph>
    <Paragraph position="3"> to truth values) define potential referent sets of tuples that consist of one entity and one situation:</Paragraph>
    <Paragraph position="5"> tation can be extended to treat common nouns as situationally-dependent properties as well, in order to handle sets like 'bachelors' that change their membership over time.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Sharing referents across
</SectionTitle>
    <Paragraph position="0"> interpretations Any method for using the environment to guide the interpretation of natural language sentences requires a tractable representation of the many possible interpretations of each input. The representation described here is based on the polynomial-sized chart produced by any dynamic programming recognition algorithm.</Paragraph>
    <Paragraph position="1"> A record of the derivation paths in any dynamic programming recognition algorithm (such as CKY (Cocke and Schwartz, 1970; Kasami, 1965; Younger, 1967) or Earley (Earley, 1970)) can be interpreted as a polynomial sized and-or graph with space complexity equal to the time complexity of recognition, whose disjunctive nodes represent possible constituents in the analysis, and whose conjunctive nodes represent binary applications of rules in the grammar. This is called a shared forest of parse trees, because it can represent an exponential number of possible parses using a polynomial number of nodes which are shared between alternative analyses (Tomita, 1985; Billot and Lang, 1989), and can be constructed and traversed in time of the same complexity (e.g. a0a2a1a4a3 a5a15a7 for context free grammars). For example, the two parse trees for the noun phrase 'button on handle beside adapter' shown in Figure 1 can be merged into the single shared forest in Figure 2 without any loss of information. These shared syntactic structures can further be associated with compositional semantic functions that correspond to the syntactic elements in the forest, to create a shared forest of trees each representing a complete expression in some logical form. This extended sharing is similar to the 'packing' approach employed in the Core Language Engine (Alshawi, 1992), except that the CLE relies on a quasi-logical form to underspecify semantic information such as quantifier scope (the calculation of which is deferred until syntactic ambiguities have been at least partially resolved by other means); whereas the approach described here extends structure sharing to incorporate a certain amount of quantifier scope ambiguity in order to allow a complete evaluation of all subderivations in a shared forest before making any disambiguation decisions in syntax.3 Various synchronous formalisms have been introduced for associating syntactic representations with logical functions in isomorphic or locally non-isomorphic derivations, including Categorial Grammars (CGs) (Wood, 1993), Synchronous Tree Adjoining Grammars (TAGs) (Joshi, 1985; Shieber and Schabes, 1990; Shieber, 1994), and Synchronous Description Tree Grammars (DTGs) (Rambow et al., 1995; Rambow and Satta, 1996). Most of these formalisms can be extended to define semantic associations over entire shared forests, rather than merely over individual parse trees, in a straightforward manner, preserving the ambiguity of the syntactic forest without exceeding its polynomial size, or the polynomial time complexity of creating or traversing it.</Paragraph>
    <Paragraph position="2"> Since one of the goals of this architecture is to use the system's representation of its environment to resolve ambiguity in its instructions, a space-efficient shared forest of logical functions will not be enough. The system must also be able to efficiently calculate the sets of potential referents in the environment for every subexpression in this forest. Fortunately, since the logical function forest shares structure between alternative analyses, many of the sets of potential referents can be shared between analyses during evaluation as well. This has the effect of building a third shared forest of potential referent sets (another and-or graph, isomorphic to the logical function forest and with the same polynomial complexity), where every conjunctive node represents the results of applying a logical function to the elements in that node's child sets, and every disjunctive node represents the union of all the potential referents in that node's child sets. The presence or absence of these environment referents at various nodes in the shared forest can be used to choose a viable parse tree from the forest, or to evaluate the truth or falsity of the input sentence without disambiguating it (by checking the presence or lack of referents at the root of the forest).</Paragraph>
    <Paragraph position="3"> For example, the noun phrase 'button on handle beside adapter' has at least two possible interpretations, represented by the two trees in Figure 1: one in which a button is on a handle and 3A similar basis on (at least partially) disambiguated syntactic representations makes similar underspecified semantic representations such as hole semantics (Bos, 1995) ill-suited for environment-based syntactic disambiguation.</Paragraph>
    <Paragraph position="5"/>
    <Paragraph position="7"> egories, and the potential environment referents are annotated just below the semantic functions in the figure. Because there are no handles next to adapters in the environment (only buttons next to adapters), the first interpretation has no environment referents at its root, so this analysis is dispreferred if it occurs within the analysis of a larger sentence. The second interpretation does have potential environment referents all the way up to the root (there is a button on a handle which is also beside an adapter), so this analysis is preferred if it occurs within the analysis of a larger sentence.</Paragraph>
    <Paragraph position="8"> The shared forest representation effectively merges the enumerated set of parse trees into a single data structure, and unions the referent sets of the nodes in these trees that have the same label and cover the same span in the string yield (such as the root node, leaves, and the PP covering 'beside adapter' in the examples above). The referent-annotated forest for this sentence therefore looks like the forest in Figure 2, in which the sets of buttons, handles, and adapters, as well as the set of things beside adapters, are shared between the two alternative interpretations. If there is a button next to an adapter, but no handle next to an adapter, the tree representing 'handle beside adapter' as a constituent may be dispreferred in disambiguation, but the NP constituent at the root is still preferred because it has potential referents in the environment due to the other interpretation.</Paragraph>
    <Paragraph position="9"> The logical function at each node is defined over the referent sets of that node's immediate children. Nodes that represent the attachment of a modifier with referent set a0 to a relation with referent set a1 produce referent sets of the form:</Paragraph>
    <Paragraph position="11"> Nodes in a logical function forest that represent the attachment of an argument with referent set a11 to a relation with referent set a1 produce referent sets of the form:</Paragraph>
    <Paragraph position="13"> effectively stripping off one of the objects in each tuple if the object is also found in the set of referents for the argument.4 This is a direct application of standard type theory to the calculation of ref4In order to show where the referents came from, the tuple objects are not stripped off in Figures 1 and 2. Instead, an additional bar is added to the function name to designate the effective last object in each tuple: the tuple a15a17a16a19a18a21a20a23a22a24a18a26a25 referenced by a16a28a27a30a29a28a31a33a32a34a27a21a35 has a22a24a18 as the last element, but the tuple referenced by a16a28a27a30a29a28a31a33a32a34a27a21a35a35 actually has a16a30a18 as the last element since the complement a22a36a18 has been already been attached.</Paragraph>
    <Paragraph position="15"> erent sets: modifiers take and return functions of the same type, and arguments must satisfy one of the input types of an applied function.</Paragraph>
    <Paragraph position="16"> Since both of these 'referent set composition' operations at the conjunctive nodes - as well as the union operation at the disjunctive nodes - are linear in space and time on the number of elements in each of the composed sets (assuming the sets are sorted in advance and remain so), the calculation of referent sets only adds a factor of a4a7 a4 to the size complexity of the forest and the time complexity of processing it, where a4a7 a4 is the number of objects and events in the run-time environment. Thus, the total space and time complexity of the above algorithm (on a context-free forest) is</Paragraph>
    <Paragraph position="18"> plexity of referent set composition will be limited by the least efficient operation.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Temporal referents
</SectionTitle>
      <Paragraph position="0"> Since the referent sets for situations are also well defined under type theory, this environment-based approach can also resolve attachment ambiguities involving verbs and verb phrases in addition to those involving only nominal referents. For example, if the interpreter is given the sentence &amp;quot;Coolant drained after test at 3:00,&amp;quot; which could mean the draining was at 3:00 or the test was at 3:00, the referents for the draining process and the testing process can be treated as time intervals in the environment history.5 First, a forest is constructed which shares the subtrees for &amp;quot;the test&amp;quot; and &amp;quot;after 3:00,&amp;quot; and the corresponding sets of referents. Each node in this forest (shown in Figure 3) is then annotated with the set of objects and intervals that it could refer to in the environment.</Paragraph>
      <Paragraph position="1"> Since there were no testing intervals at 3:00 in the environment, the referent set for the NP 'test after 3:00' is evaluated to the null set. But since there is an interval corresponding to a draining process (a23a24a12 ) at the root, the whole VP will still be preferred as constituent due to the other interpretation. null</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Quantifier scoping
</SectionTitle>
      <Paragraph position="0"> The evaluation of referents for quantifiers also presents a tractability problem, because the functions they correspond to in the Montague analysis map two sets of entities to a truth value. This means that a straightforward representation of the potential referents of a quantifier such as 'at least one' would contain every pair of non-empty sub-sets of the set a7 of all entities, with a cardinality on the order of a8 a17a10a9a11a12a9 . If the evaluation of referents is deferred until quantifiers are composed with the common nouns they quantify over, the 5The composition of time intervals, as well as spatial regions and other types of situational referents, is more complex than that outlined for objects, but space does not permit a complete explanation.</Paragraph>
      <Paragraph position="1"> input sets would still be as large as the power sets of the nouns' potential referents. Only if the evaluation of referents is deferred until complete NPs are composed as arguments (as subjects or objects of verbs, for example) can the output sets be restricted to a tractable size.</Paragraph>
      <Paragraph position="2"> This provision only covers in situ quantifier scopings, however. In order to model raised scopings, arbitrarily long chains of raised quantifiers (if there are more than one) would have to be evaluated before they are attached to the verb, as they are in a CCG-style function composition analysis of raising (Park, 1996).6 Fortunately, universal quantifiers like 'each' and 'every' only choose the one maximal set of referents out of all the possible subsets in the power set, so any number of raised universal quantifier functions can be composed into a single function whose referent set would be no larger than the set of all possible entities. null It may not be possible to evaluate the potential referents of non-universal raised quantifiers in polynomial time, because the number of potential subsets they take as input is on the order of the power set of the noun's potential referents. This apparent failure may hold some explanatory power, however, since raised quantifiers other than 'each' and 'every' seem to be exceedingly rare in the data. This scarcity may be a result of the significant computational complexity of evaluating them in isolation (before they are composed with a verb).</Paragraph>
    </Section>
  </Section>
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