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<Paper uid="P96-1013">
  <Title>Parsing for Semidirectional Lambek Grammar is NP-Complete</Title>
  <Section position="4" start_page="99" end_page="99" type="concl">
    <SectionTitle>
4 Conclusion
</SectionTitle>
    <Paragraph position="0"> We have defined a variant of Lambek's original calculus of types that allows abstracted-over categories to freely permute. Grammars based on SOl- can generate any context-free language and more than that. The parsing problem for SD\[, however, we have shown to be NP-complete. This result indicates that efficient parsing for grammars that allow for large numbers of unbounded dependencies from within one node may be problematic, even in the categorial framework. Note that the fact, that this problematic case doesn't show up in the correct analysis of normal NL sentences, doesn't mean that a parser wouldn't have to try it, unless some arbitrary bound to that number is assumed. For practical grammar engineering one can devise the motto avoid accumulation of unbounded dependencies by whatever means.</Paragraph>
    <Paragraph position="1"> On the theoretical side we think that this result for S01 is also of some importance, since SDI_ exhibits a core of logical behaviour that any (Lambek-based) logic must have which accounts for non-peripheral extraction by some form of permutation. And hence, this result increases our understanding of the necessary computational properties of such richer systems. To our knowledge the question, whether the Lambek calculus itself or its associated parsing problem are NP-hard, are still open.</Paragraph>
  </Section>
class="xml-element"></Paper>