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<Paper uid="C90-2041">
  <Title>THE COMPLEXITY OF PARSING WITH EXTENDED CATEGOP IAL GP AMMARS</Title>
  <Section position="6" start_page="0" end_page="0" type="concl">
    <SectionTitle>
4 Conclusion
</SectionTitle>
    <Paragraph position="0"> We have presented an extension of tile CKY-algorithm for an instance of the family of extended categorial grammars which uses hypothetical reasoning, i.e. for the bidirectional Lambek categorial grammars. Although the current algorithm has an undesirable time complexity in its gener~d case, one carl define restricted versions which cover tile kind of structures found in linguistic examples with time complexity O(n 3) or O(n 4) depending on the. fact  whether one wishes to describe single or double extraction from phrases. It is an interesting question whether better Mgorithms for the general case can be found.</Paragraph>
    <Paragraph position="1"> There are many ways to improve this new algorithm concerning its &amp;quot;average complexity&amp;quot; in practical applications. For example, the search for a partner item in a completer-step could be reduced by using appropriate heuristics. Further incre~es in efficiency can be expected, if one allows for arbitrary embedding of disjunctions and slashes in categories (of. \[Benthem 19891, \[Morrill 1989\]) instead of using the flat set notation which amounts to a disjunctive normal form of categories in the lexicon.</Paragraph>
    <Paragraph position="2"> Theoretical complexity results for a group of categorial grammars reaching beyond context-free languages are known, as well. From the fact that certain extended versions of categorial grammars (CCG) are equivalent with the Tree Adjoining Grammars (TAG) (\[Weir, Josh\[ 19881), the O(n 4 log n)time complexity of TAG-parsers (\[tIarbusch 1989\]), in principle, carries over to CCG.</Paragraph>
    <Paragraph position="3"> The connection of the propositional grammar calculi as discussed above with a unification device for first order germs is done by taking care of the variable bindings in the hypothetical categories which are asserted by the abstraction rules. The recognition problem is still decidable (cf. \[Pareschi 1988\]) although surely not well-behaved as this is the case for all non-propositional grammars. For example, the recognition problem for a. basic categorial grammar witti feature unification is NP-complete (\[Morrill 1988\]). Higher order versions of categorial grammars like the ones being produced in the CUG/UCG-frameworks (\[Uszkoreit 1986\],\[Zeevat et at. 1986\]) where variables can range over categories are in general undecidable because higher order unification itself is undecidable.</Paragraph>
  </Section>
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