k-Valued Non-associative Lambek Grammars (Without Product) Form a Strict Hierarchy of Languages

  • Denis Béchet
  • Annie Foret
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3492)


The notion of k-valued categorial grammars where a word is associated to at most k types is often used in the field of lexicalized grammars as a fruitful constraint for obtaining several properties like the existence of learning algorithms. This principle is relevant only when the classes of k-valued grammars correspond to a real hierarchy of languages. This paper establishes the relevance of this notion for two related grammatical systems. In the first part, the classes of k-valued non-associative Lambek (NL) grammars without product is proved to define a strict hierarchy of languages. The second part introduces the notion of generalized functor argument for non-associative Lambek (NL  ∅ ) calculus without product but allowing empty antecedent and establishes also that the classes of k-valued (NL  ∅ ) grammars without product form a strict hierarchy of languages.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bar-Hillel, Y.: A quasi arithmetical notation for syntactic description. Language 29, 47–58 (1953)CrossRefGoogle Scholar
  2. 2.
    Lambek, J.: The mathematics of sentence structure. American mathematical monthly 65, 154–169 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Joshi, A.K., Shabes, Y.: Tree-adjoining grammars and lexicalized grammars. In: Tree Automata and LGS, Elsevier Science, Amsterdam (1992)Google Scholar
  4. 4.
    Gold, E.: Language identification in the limit. Information and control 10, 447–474 (1967)zbMATHCrossRefGoogle Scholar
  5. 5.
    Kanazawa, M.: Learnable Classes of Categorial Grammars. Studies in Logic, Language and Information. Center for the Study of Language and Information (CSLI) and The European association for Logic, Language and Information (FOLLI), Stanford, California (1998)zbMATHGoogle Scholar
  6. 6.
    Béchet, D., Foret, A.: k-valued non-associative lambek grammars are learnable from function-argument structures. In: de Queiroz, R., Pimentel, E., Figueiredo, L. (eds.) Electronic Notes in Theoretical Computer Science, vol. 84, pp. 1–13. Elsevier, Amsterdam (2003)Google Scholar
  7. 7.
    Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of language and its mathematical aspects, pp. 166–178. American Mathematical Society (1961)Google Scholar
  8. 8.
    Kandulski, M.: The non-associative lambek calculus. In: Buszkowski, W., Marciszewski, W., Van Bentem, J. (eds.) Categorial Grammar, pp. 141–152. Benjamins, Amsterdam (1988)Google Scholar
  9. 9.
    Aarts, E., Trautwein, K.: Non-associative Lambek categorial grammar in polynomial time. Mathematical Logic Quaterly 41, 476–484 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Buszkowski, W.: Mathematical linguistics and proof theory. In: [14], ch. 12, pp. 683–736Google Scholar
  11. 11.
    Moortgat, M.: Categorial type logic. In: [14], ch. 2, pp. 93–177.Google Scholar
  12. 12.
    de Groote, P.: Non-associative Lambek calculus in polynomial time. In: TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 128–139. Springer-Verlag, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    de Groote, P., Lamarche, F.: Classical non-associative lambek calculus. Studia Logica 71(3), 355–388 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    van Benthem, J., ter Meulen, A.: Handbook of Logic and Language. North-Holland Elsevier, Amsterdam (1997)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Denis Béchet
    • 1
  • Annie Foret
    • 2
  1. 1.LINA – FRE 2729Université de Nantes & CNRSNantes Cedex 03France
  2. 2.IRISAUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations