Advertisement

Lambek Grammars with the Unit

  • Stepan Kuznetsov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7395)

Abstract

Pentus’ theorem states that any language generated by a Lambek grammar is context-free. We present a substitution that reduces the Lambek calculus enriched with the unit constant to the variant of the Lambek calculus that does not contain the unit (but still allows empty premises), and use this substitution to prove that any language generated by a categorial grammar based on the Lambek calculus with the unit is context-free.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bar-Hillel, Y., Gaifman, C., Shamir, E.: On categorial and phrase-structure grammars. Bull. Res. Council Israel Sect. F 9F, 1–16 (1960)MathSciNetGoogle Scholar
  2. Buszkowski, W.: The equivalence of unidirectional Lambek categorial grammars and context-free grammars. Zeitschr. für math. Logik und Grundl. der Math. 31, 369–384 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Kuznetsov, S.: Lambek grammars with one division and one primitive type. Unpublished manuscript (2010)Google Scholar
  4. Lambek, J.: The mathematics of sentence structure. American Math. Monthly 65(3), 154–170 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Lambek, J.: Deductive systems and categories II: Standard constructions and closed categories. In: Hilton, P. (ed.) Category Theory, Homology Theory and Their Applications I. Lect. Notes Math., vol. 86, pp. 76–122. Springer, Berlin (1969)CrossRefGoogle Scholar
  6. Métayer, F.: Polynomial equivalence among systems LLNC, LLNCa and LLNC0. Theor. Comput. Sci. 227(1), 221–229 (1999)zbMATHCrossRefGoogle Scholar
  7. Pentus, M.: Lambek grammars are context free. In: Proc. of the 8th Annual IEEE Symposium on Logic in Computer Science, pp. 429–433. IEEE Computer Society Press, Los Alamitos (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stepan Kuznetsov
    • 1
  1. 1.Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and MathematicsMoscow State UniversityRussia

Personalised recommendations