The Generative Capacity of the Lambek–Grishin Calculus: A New Lower Bound

  • Matthijs Melissen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5591)


The Lambek–Grishin calculus LG is a categorial type logic obtained by adding a family of connectives Open image in new window dual to the family { ⊗ , /, \}, and adding interaction postulates between the two families of connectives thus obtained. In this paper, we prove a new lower bound on the generative capacity of LG, namely the class of languages that are the intersection of a context-free language and the permutation closure of a context-free language. This implies that LG recognizes languages like the MIX language, e.g. the permutation closure of Open image in new window , and Open image in new window , which can not be recognized by tree adjoining grammars.


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  1. 1.
    Bar-Hillel, Y.: A quasi-arithmetical notation for syntactic description. Language 29, 47–58 (1953)CrossRefzbMATHGoogle Scholar
  2. 2.
    Boullier, P.: Range concatenation grammars. In: New Developments in Parsing Technology, pp. 269–289. Kluwer Academic Publishers, Norwell (2004)CrossRefGoogle Scholar
  3. 3.
    Buszkowski, W.: Generative capacity of nonassociative Lambek calculus. Bulletin of Polish Academy of Sciences: Mathematics 34, 507–516 (1986)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Castaño, J.M.: Global index grammars and descriptive power. J. of Logic, Lang. and Inf. 13(4), 403–419 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grishin, V.N.: On a generalization of the Ajdukiewicz-Lambek system. In: Mikhailov, A.I. (ed.) Studies in Non-classical Logics and Formal Systems, pp. 315–343. Nauka, Moscow (1983)Google Scholar
  6. 6.
    Huybregts, R.: The weak inadequacy of context-free phrase structure grammars. In: Trommelen, M., de Haan, G.J., Zonneveld, W. (eds.) Van Periferie Naar Kern, pp. 81–99. Foris Publications, Dordrecht (1984)Google Scholar
  7. 7.
    Joshi, A., Levy, L.S., Takahashi, M.: Tree adjunct grammars. Journal of Computer and System Sciences 10(1), 136–163 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Joshi, A., Schabes, Y.: Tree-adjoining grammars. In: Handbook of Formal Languages. Beyond Words, vol. 3, pp. 69–123. Springer, Inc., New York (1997)CrossRefGoogle Scholar
  9. 9.
    Kandulski, M.: The equivalence of nonassociative Lambek categorial grammars and context-free grammars. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 34(1), 41–52 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lambek, J.: On the calculus of syntactic types. In: Jacobsen, R. (ed.) Structure of Language and its Mathematical Aspects. Proceedings of Symposia in Applied Mathematics, vol. XII, pp. 166–178. American Mathematical Society, Providence (1961)CrossRefGoogle Scholar
  11. 11.
    Melissen, M.: Lambek–Grishin calculus extended to connectives of arbitrary arity. In: Proceedings of the 20th Belgian-Netherlands Conference on Artificial Intelligence (2008)Google Scholar
  12. 12.
    Moortgat, M.: Symmetries in natural language syntax and semantics: The lambek-grishin calculus. In: Leivant, D., de Queiroz, R. (eds.) WoLLIC 2007. LNCS, vol. 4576, pp. 264–284. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Moot, R.: Lambek grammars, tree adjoining grammars and hyperedge replacement grammars. In: Proceedings of the TAG+ Conference, HAL - CCSD (2008)Google Scholar
  14. 14.
    Parikh, R.: On context-free languages. J. ACM 13(4), 570–581 (1966)CrossRefzbMATHGoogle Scholar
  15. 15.
    Pentus, M.: Lambek grammars are context free. In: Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science, pp. 429–433. IEEE Computer Society Press, Los Alamitos (1993)Google Scholar
  16. 16.
    Sipser, M.: Introduction to the Theory of Computation. Course Technology (December 1996)Google Scholar
  17. 17.
    Van Benthem, J.: Language in action: categories, lambdas and dynamic logic. Studies in logic and the foundations of mathematics, vol. 130. North-Holland, Amsterdam (1991)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matthijs Melissen
    • 1
    • 2
  1. 1.Universiteit UtrechtThe Netherlands
  2. 2.Université de LuxembourgLuxembourg

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