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<Paper uid="P01-1011">
  <Title>Underspecified Beta Reduction</Title>
  <Section position="7" start_page="0" end_page="0" type="concl">
    <SectionTitle>
7 Conclusion
</SectionTitle>
    <Paragraph position="0"> In this paper, we have shown how to perform an underspecifieda0 -reduction operation in the CLLS framework. This operation transforms underspecified descriptions of higher-order formulas into descriptions of their a0 -reducts. It can be used to essentially a0 -reduce all readings of an ambiguous sentence at once.</Paragraph>
    <Paragraph position="1"> It is interesting to observe how our under-specified a0 -reduction interacts with parallelism constraints that were introduced to model ellipses. Consider the elliptical three-reading example &amp;quot;Peter sees a loophole. Every lawyer does too.&amp;quot; Under the standard analysis of ellipsis in CLLS (Egg et al., 2001), &amp;quot;Peter&amp;quot; must be represented as a generalized quantifier to obtain all three readings. This leads to a spurious ambigu- null ity in the source sentence, which one would like to get rid of by a0 -reducing the source sentence.</Paragraph>
    <Paragraph position="2"> Our approach can achieve this goal: Adding a0 -reduction constraints for the source sentence leaves the original copy intact, and the target sentence still contains the ambiguity.</Paragraph>
    <Paragraph position="3"> Under the simplifying assumption that all redexes are linear, we can show that it takes time</Paragraph>
    <Paragraph position="5"> a56 to perform a9 steps of underspecified a0 -reduction on a constraint with a98 variables. This is feasible for large a9 as long as a98a12a11a14a13 a85 , which should be sufficient for most reasonable sentences. If there are non-linear redexes, the present algorithm can take exponential time because subterms are duplicated. The same problem is known in ordinary a1 -calculus; an interesting question to pursue is whether the sharing techniques developed there (Lamping, 1990) carry over to the underspecification setting.</Paragraph>
    <Paragraph position="6"> In Sec. 6, we only employ propagation rules; that is, we never disambiguate. This is conceptually very nice, but on more complex examples (e.g. in many cases with nonlinear redexes) disambiguation is still needed.</Paragraph>
    <Paragraph position="7"> This raises both theoretical and practical issues.</Paragraph>
    <Paragraph position="8"> On the theoretical level, the questions of completeness (elimination of all redexes) and confluence still have to be resolved. To that end, we first have to find suitable notions of completeness and confluence in our setting. Also we would like to handle larger classes of examples without disambiguation. On the practical side, we intend to implement the procedure and disambiguate in a controlled fashion so we can reduce completely and still disambiguate as little as possible.</Paragraph>
  </Section>
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