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<Paper uid="P03-1045">
  <Title>k-valued Non-Associative Lambek Categorial Grammars are not Learnable from Strings</Title>
  <Section position="5" start_page="0" end_page="0" type="concl">
    <SectionTitle>
4 Conclusion and Remarks
</SectionTitle>
    <Paragraph position="0"> Lambek grammars. We have shown that without empty sequence, non-associative Lambek rigid grammars are not learnable from strings. With this result, the whole landscape of Lambek-like rigid grammars (or k-valued for an arbitrary k) is now described as for the learnability question (from strings, in Gold's model).</Paragraph>
    <Paragraph position="1"> Non-learnability for subclasses. Our construct is of order 5 and does not use the product operator.</Paragraph>
    <Paragraph position="2"> Thus, we have the following corollaries: Restricted connectives: k-valued NL, NL;, L and L; grammars without product are not learnable from strings.</Paragraph>
    <Paragraph position="3"> Restricted type order: - k-valued NL, NL;, L and L; grammars (without product) with types not greater than order 5 are not learnable from strings4.</Paragraph>
    <Paragraph position="4"> - k-valued free pregroup grammars with types not greater than order 1 are not learnable from strings5.</Paragraph>
    <Paragraph position="5"> The learnability question may still be raised for NL grammars of order lower than 5.</Paragraph>
    <Paragraph position="6">  Special learnable subclasses. Note that however, we get specific learnable subclasses of k-valued grammars when we consider NL, NL;, L or L; without product and we bind the order of types in grammars to be not greater than 1. This holds for all variants of Lambek grammars as a corollary of the equivalence between generation in classical categorial grammars and in Lambek systems for grammars with such product-free types (Buszkowski, 2001). Restriction on types. An interesting perspective for learnability results might be to introduce reasonable restrictions on types. From what we have seen, the order of type alone (order 1 excepted) does not seem to be an appropriate measure in that context. Structured examples. These results also indicate the necessity of using structured examples as input of learning algorithms. What intermediate structure should then be taken as a good alternative between insufficient structures (strings) and linguistic unrealistic structures (full proof tree structures) remains an interesting challenge.</Paragraph>
  </Section>
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