Studia Logica

, Volume 71, Issue 3, pp 355–388 | Cite as

Classical Non-Associative Lambek Calculus

  • Philippe de Groote
  • François Lamarche


We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

non-associative Lambek calculus linear logic proof-net 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aarts, E., and K. Trautwein, ‘Non-associative Lambek categorial grammar in polynomial time’, Mathematical Logic Quaterly 41:476-484, 1995.Google Scholar
  2. [2]
    Abrusci, M., ‘Phase semantics and sequent calculus for pure non-commutative classical linear logic’, Journal of Symbolic Logic 56(4):1403-1451, 1991.Google Scholar
  3. [3]
    Abrusci, M., and E. Maringelli, ‘A new Correctness Criterion fo Cyclic Multiplicative proof-nets’, Journal of Logic, Language and Information 7(4):449-459, 1998.Google Scholar
  4. [4]
    Abrusci, M., and P. Ruet, ‘Non commutative logic I: the multiplicative fragment’, Annals of Pure and Applied Logic 101(1):29-64, 2000.Google Scholar
  5. [5]
    Abrusci, V. M., ‘Non-commutative proof-nets’, in J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, London Mathematical Society Lecture Notes, pages 271-296. Cambridge University Press, 1995.Google Scholar
  6. [6]
    Berge, C., Graphs, North-Holland, second revised edition edition, 1985.Google Scholar
  7. [7]
    Buszkowski, W., Logical Foundations of Ajdukiewicz-Lambek Categorial Grammars, Polish Scientific Publishers, 1989 (in Polish).Google Scholar
  8. [8]
    Buszkowski, W., ‘Generative power of non-associative Lambek Calculus’, Bull. Polish Acad. Sci. Math. 34:507-516, 1986.Google Scholar
  9. [9]
    de Groote, Ph., ‘A dynamic programming approach to categorial deduction’, in H. Ganzinger, editor, 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, pages 1-15. Springer Verlag, 1999.Google Scholar
  10. [10]
    Fleury, A., La règle d'échange: logique lineaire multiplicative tressée, Thèse de Doctorat, spécialité Mathématiques, Université Paris 7, 1996.Google Scholar
  11. [11]
    Kandulski, M., ‘The equivalence of nonassociative Lambek categorial grammars and context-free grammars’, Z. Math. Logik Grundlag. Math. 34:103-114, 1988.Google Scholar
  12. [12]
    Girard, J.-Y., ‘Linear logic’, Theoretical Computer Science 50:1-102, 1987.Google Scholar
  13. [13]
    Lafont, Y., ‘Interaction nets’, in Proceedings of the 17th ACM symposium on Principles of Programming Languages, ACM Press, 1990.Google Scholar
  14. [14]
    Lamarche, F., and C. Retoré, ‘Proof nets for the Lambek calculus’, in M. Abrusci and C. Casadio, editors, Proofs and Linguistic Categories, Proceedings 1996 Roma Workshop. Cooperativa Libraria Universitaria Editrice Bologna, 1996.Google Scholar
  15. [15]
    Lambek, J., ‘The mathematics of sentence structure’, Amer. Math. Monthly, 65:154-170, 1958.Google Scholar
  16. [16]
    Lambek, J., ‘On the calculus of syntactic types’, in R. Jakobson, editor, Structure of Language and its Mathematical Aspects, pages 166-178, Providence, 1961.Google Scholar
  17. [17]
    Moortgat, M., ‘Categorial type logic’, in J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, chapter 2. Elsevier, 1997.Google Scholar
  18. [18]
    Puite, Q., and R. Moot, ‘Proof nets for the multimodal Lambek calculus’, Preprint nr. 1096, Department of Mathematics, University of Utrecht, March 1999.Google Scholar
  19. [19]
    Moortgat, M., and R. Oehrle, ‘Proof nets for the grammatical base logic’, in V. M. Abrusci, C. Casadio, and G. Sandri, editors, Proceedings of the IV Roma Workshop, October 1997. Cooperativa Libraria Universitaria Editrice Bologna, 1999.Google Scholar
  20. [20]
    Morrill, G., ‘Memoisation of categorial proof nets: parallelism in categorial processing’, in V. M. Abrusci and C. Casadio, editors, Proofs and Linguistic Categories, Proceedings 1996 Roma Workshop. Cooperativa Libraria Universitaria Editrice Bologna, 1996.Google Scholar
  21. [21]
    Nagayama, M., and M. Okada, ‘A graph-theoretical characterization theorem for multiplicative fragment of non-commutative linear logic’, in J.-Y. Girard, M. Okada, and A. Scedrov, editors, Linear'96, Proceedings of the Tokyo Meeting, volume 3 of Electronic Notes in Theoretical Computer Science. Elsevier-North-Holland, 1996.Google Scholar
  22. [22]
    Retoré, C., ‘Calcul de Lambek et logique linéaire’, Traitement Automatique des Langues 37(2):39-70, 1997.Google Scholar
  23. [23]
    Roorda, D., Resource Logics: proof-theoretical investigations. PhD thesis, University of Amsterdam, 1991.Google Scholar
  24. [24]
    Ruet, P., Logique non-commutative et programmation concurrente par contraintes, Thèse de Doctorat, logique et fondement de l'informatique, Université Denis Diderot, Paris 7, 1997.Google Scholar
  25. [25]
    Ruet, P., ‘Non commutative logic II: Sequent calculus and phase semantics’, Math. Struc. Comp. Sci. 10(2):277-312.Google Scholar
  26. [26]
    Szczerba, M., ‘Representation theorems for residuated groupoids’, Proceedings of LACL96, September 1996, Nancy. Springer Lecture Notes in Artificial Intelligence, vol. 1328.Google Scholar
  27. [27]
    van Benthem, J., Language in Action: Categories, Lambdas and Dynamic Logic, volume 130 of Sudies in Logic and the foundation of mathematics. North-Holland, Amsterdam, 1991.Google Scholar
  28. [28]
    Yetter, D. N., ‘Quantales and (non-commutative) linear logic’, Journal of Symbolic Logic 55:41-64, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Philippe de Groote
    • 1
  • François Lamarche
    • 1
  1. 1.Projet CalligrammeLORIA UMR no 7503 — INRIA, Campus ScientifiqueVandœuvre lès Nancy CedexFrance

Personalised recommendations