Studia Logica

, Volume 87, Issue 2–3, pp 323–342

On the Logic of β-pregroups

Article

Abstract

In this paper we concentrate mainly on the notion of β-pregroups, which are pregroups (first introduced by Lambek [18] in 1999) enriched with modality operators. β-pregroups were first proposed by Fadda [11] in 2001. The motivation to introduce them was to limit (locally) the associativity in the calculus considered. In this paper we present this new calculus in the form of a rewriting system, prove the very important feature of this system - that in a given derivation the non-expanding rules must always proceed non-contracting ones in order the derivation to be minimal (normalization theorem). We also propose a sequent system for this calculus and prove the cut elimination theorem for it. As an illustration we show how to use β-pregroups for linguistical applications.

Keywords

Pregroup β-pregroup normalization theorem cut elimination 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrusci, V.M., ‘Lambek Calculus, Cyclic Multiplicative -Additive Linear Logic, Noncommutative Multiplicative - Additive Linear Logic: language and sequent calculus’, in V.M. Abrusci and C. Casadio (eds.), Proofs and Linguistic Categories, Proceedings 1996 Roma Workshop, Bologna, 1996, pp. 21–48.Google Scholar
  2. 2.
    Ajdukiewicz K. (1935). ‘Die syntaktische Konnexität’. Studia Philosophica 1:1–27Google Scholar
  3. 3.
    Buszkowski W. (1986). ‘Generative capacity of nonassociative Lambek calculus’. Bull. Polish Academy Scie. Math. 34:507–516Google Scholar
  4. 4.
    Buszkowski, W., Logiczne podstawy gramatyk kategorialnych Ajdukiewicza-Lambeka, PWN, Warszawa, 1989.Google Scholar
  5. 5.
    Buszkowski W. (1996). ‘Extending Lambek grammars to basic categorial grammars’. Journal of Logic, Language and Information 5:279–295CrossRefGoogle Scholar
  6. 6.
    Buszkowski, W., ‘Mathematical linguistics and proof theory’, in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, MIT Press, Cambridge Mass., 1997, pp. 683–736.Google Scholar
  7. 7.
    Buszkowski, W., ‘Lambek Grammars Based on Pregroups’, in Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001, pp. 95–109.Google Scholar
  8. 8.
    Buszkowski, W., ‘Cut elimination for the Lambek calculus of adjoints’, in V.M. Abrusci, C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, ISBN 88-8319-747-8, Bulzoni Editore, Roma, 2002, pp. 85–94.Google Scholar
  9. 9.
    Buszkowski W. (2003). ‘Sequent systems for compact bilinear logic’. Mathematical Logic Quarterly 49(5):467–474CrossRefGoogle Scholar
  10. 10.
    Casadio, C., and J. Lambek, ‘An Algebraic Analysis of Clitic Pronouns in Italian’, in Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001, pp. 110– 124.Google Scholar
  11. 11.
    Fadda M., ‘Towards flexible pregroup grammars’, in V.M. Abrusci and C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, ISBN 88-8319-747-8, Bulzoni Editore, Roma, 2002, pp. 95–112.Google Scholar
  12. 12.
    Fuchs L. (1963). Partially Ordered Algebraic Systems. Pergamon Press, OxfordGoogle Scholar
  13. 13.
    Kiślak, A., ‘Parsing based on pregroups. Comments on the Lambek theory of syntactic structure’, Formal Methods and Intelligent Techniques in Control, Decision Making, Multimedia and Robotics, Warsaw, 2000, pp. 41–49.Google Scholar
  14. 14.
    Kiślak, A., ‘Pregroups versus English and Polish grammar’, in V.M. Abrusci, and C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, ISBN 88-8319- 747-8, Bulzoni Editore, Roma, 2002, pp. 129–154.Google Scholar
  15. 15.
    Kiślak-Malinowska A. (2004). ‘Conjoinability in pregroups’. Fundamenta Informaticae 61(1):29–36Google Scholar
  16. 16.
    Kiślak-Malinowska, A., ’Pregroups as a tool for typing relative pronouns in Polish’, Proceedings of Categorial Grammars, An efficient tool for Natural Language Processing, Montpellier, 2004, pp. 114–128.Google Scholar
  17. 17.
    Lambek J. (1958). ‘The mathematics of sentence structure’. The American Mathematical Monthly 65:154–170CrossRefGoogle Scholar
  18. 18.
    Lambek, J., ‘Type grammars revisited’, in A. Lecomte, F. Lamarche and G. Perrier, Logical Aspects of Computational Linguistics, LNAI 1582, Springer, Berlin, 1999, pp. 1–27. ipt.Google Scholar
  19. 19.
    Lambek J. (2001). ‘Type Grammars as Pregroups’. Grammars 4:21–39CrossRefGoogle Scholar
  20. 20.
    Lambek, J., ‘Pregroups: a new algebraic approach to sentence structure’ in V.M. Abrusci, C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, ISBN 88-8319-747-8, Bulzoni Editore, Roma, 2002, pp. 39–54.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Logic and Computer ScienceUniversity of Warmia and MazuryOlsztynPoland

Personalised recommendations