Skip to main content
SpringerLink
Log in

On the Logic of β-pregroups

  • Published:
Studia Logica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

  2. Ajdukiewicz K. (1935). ‘Die syntaktische Konnexität’. Studia Philosophica 1:1–27

    Google Scholar 

  3. Buszkowski W. (1986). ‘Generative capacity of nonassociative Lambek calculus’. Bull. Polish Academy Scie. Math. 34:507–516

    Google Scholar 

  4. Buszkowski, W., Logiczne podstawy gramatyk kategorialnych Ajdukiewicza-Lambeka, PWN, Warszawa, 1989.

  5. Buszkowski W. (1996). ‘Extending Lambek grammars to basic categorial grammars’. Journal of Logic, Language and Information 5:279–295

    Article  Google Scholar 

  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.

  7. Buszkowski, W., ‘Lambek Grammars Based on Pregroups’, in Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001, pp. 95–109.

  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.

  9. Buszkowski W. (2003). ‘Sequent systems for compact bilinear logic’. Mathematical Logic Quarterly 49(5):467–474

    Article  Google Scholar 

  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.

  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.

  12. Fuchs L. (1963). Partially Ordered Algebraic Systems. Pergamon Press, Oxford

    Google Scholar 

  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.

  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.

  15. Kiślak-Malinowska A. (2004). ‘Conjoinability in pregroups’. Fundamenta Informaticae 61(1):29–36

    Google Scholar 

  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.

  17. Lambek J. (1958). ‘The mathematics of sentence structure’. The American Mathematical Monthly 65:154–170

    Article  Google Scholar 

  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.

  19. Lambek J. (2001). ‘Type Grammars as Pregroups’. Grammars 4:21–39

    Article  Google Scholar 

  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.

Download references

Author information

Authors and Affiliations

  1. Department of Logic and Computer Science, University of Warmia and Mazury, Żołnierska 14, Olsztyn, Poland

    Aleksandra Kiślak-Malinowska

Authors
  1. Aleksandra Kiślak-Malinowska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aleksandra Kiślak-Malinowska.

Additional information

Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kiślak-Malinowska, A. On the Logic of β-pregroups. Stud Logica 87, 323–342 (2007). https://doi.org/10.1007/s11225-007-9090-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-007-9090-5

Keywords

Search

Navigation