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<Paper uid="P89-1033">
  <Title>PARSING AS NATURAL DEDUCTION</Title>
  <Section position="7" start_page="277" end_page="277" type="concl">
    <SectionTitle>
5 Conclusion
</SectionTitle>
    <Paragraph position="0"> The cut-free and product-free part of Lambek Calculus has been augmented by certain constraints in order to yield only normal form proofs, i.e. only one proof per &amp;quot;reading&amp;quot; of a sentence. Thus, theorem provers for Larnbek Calculus become realistic tools to be employed as parsers for categorial grammar.</Paragraph>
    <Paragraph position="1"> General efficiency considerations would be of interest. Unconstrained Lambek Calculus seems to be absolutely inefficient, i.e. exponential. So far, no results are known as to how the use of the nesting constraints and the count invariant filter systematically affect the complexity. At least intuitively, it seems clear that their effects are drastic, because due to the former, considerably fewer proofs are generated at all, and due to the latter, substantially fewer irrelevant sub-proofs are pursued.</Paragraph>
    <Paragraph position="2"> From a linguistic standpoint, for example, the following questions have to be discussed: How does Lambek Calculus interact with a sophisticated lexicon containing e.g. lexical rules? Which would be linguistically desirable extensions of the inference rule system that would not throw over the properties (e.g. normal form proof) of the original Lambek Calculus? An implementation of the normal form theorem prover is currently being used for experimentation concerning these questions.</Paragraph>
  </Section>
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