Lambek Grammars with One Division Are Decidable in Polynomial Time

  • Yury Savateev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)


Lambek grammars provide a useful tool for studying formal and natural languages. The generative power of unidirectional Lambek grammars equals that of context-free grammars. However, no feasible algorithm was known for deciding membership in the corresponding formal languages. In this paper we present a polynomial algorithm for deciding whether a given word belongs to a language generated by a given unidirectional Lambek grammar.


Polynomial Time Induction Assumption Algorithm Description Angle Bracket Primitive Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aarts, E., Trautwein, K.: Non-associative Lambek categorial grammar in polynomial time. Mathematical Logic Quarterly 41, 476–484 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Buszkowski, W.: The equivalence of unidirectional Lambek categorial grammars and context-free grammars. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 31(4), 369–384 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    de Groote, P.: The non-associative Lambek calculus with product in polynomial time. In: Murray, N.V. (ed.) Automated Reasoning with Analytic Tableaux and Related Methods, pp. 128–139. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65(3), 154–170 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of Language and Its Mathematical Aspects, Proc. Symposia Appl. Math., vol. 12, pp. 166–178. Amer. Math. Soc, Providence, RI (1961)Google Scholar
  6. 6.
    Pentus, M.: Lambek calculus is NP-complete. Theoretical Computer Science 357(1–3), 186–201 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Pentus, M.: Lambek grammars are context free. In: Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science, pp. 429–433 (1993)Google Scholar
  8. 8.
    Savateev, Y.: The derivability problem for lambek calculus with one division. Technical report, Utrecht University, Artificial Intelligence Preprint Series no. 56 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yury Savateev
    • 1
  1. 1.Department of Mathematical Logic, Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations