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<Paper uid="J99-1002">
  <Title>Default Representation in Constraint-based Frameworks</Title>
  <Section position="6" start_page="98" end_page="102" type="concl">
    <SectionTitle>
7. Conclusion
</SectionTitle>
    <Paragraph position="0"> We have argued that for default unification to achieve the combination of perspicuity and declarativity familiar from normal unification, default unification should share some of its properties--such as determinacy and order independence. At the same time, default unification should respect the behavior of defaults, such as the overriding of default information by more specific conflicting defaults. We have also argued here and elsewhere (Lascarides and Copestake, in press) that some linguistic phenom- null Computational Linguistics Volume 25, Number 1 ena suggest that there are conventional default constraints that persist beyond the lexicon, and are potentially overridden by more open-ended reasoning with (default) pragmatic knowledge in a discourse context. This requires a definition of default unification where the default results of unification are marked as default, and thus distinguished from the indefeasible results. We provided a definition of default unification known as YADU, which intuitively models the incorporation of the maximal amount of default information into the result, by adapting Carpenter's (1993) version of asymmetric default unification to the situation where default and nondefault information is distinguished in a single structure and defaults may have different priorities. Our definition was formally proven to meet the above requirements. We suggested that such a definition of default unification can improve the declarativity of existing uses of default inheritance within the lexicon because it does not require one to pre-specify the order in which information is to be accumulated.</Paragraph>
    <Paragraph position="1"> Despite YADU's factorial worst-case complexity behavior, its use does not significantly decrease overall system performance when compared to a monotonic encoding for the examples we have tried in the LKB system. These results are preliminary and obviously only true relative to our particular implementation and style of grammar encoding, but they lead us to believe that the worst-case complexity behavior does not preclude the use of YADU in typed feature structure implementations. Although we only discussed a few examples in Section 4, we believe these illustrate the potential utility of defaults in a range of different contexts within a grammar and lexicon. We hope to report on a comparison between the monotonic and YADU versions of the English Resource Grammar in a later paper.</Paragraph>
    <Paragraph position="2"> Appendix Proof of Lemma 1 First we prove that Bot 12 = Bot 21. Note that by order independence of M, I12 : /21. And by order independence of set union T 1 U T 2 = T 2 U T 1. So Bot 12 =des {(F, t&gt; c T 1 U T 2 such that I12 N F = _L} = {(F,t&gt; c T2UT 1 such that I21MF = _L} =ee/ Bot 21.</Paragraph>
    <Paragraph position="3"> So:</Paragraph>
    <Paragraph position="5"> Thus we need to prove that</Paragraph>
    <Paragraph position="7"> But this follows immediately by the definition of T for BasicTDFS(I, ID) (T is the set of atomic TFSs that are subsumed by I rG ID but not subsumed by I). \[\]</Paragraph>
  </Section>
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