Studia Logica

, Volume 87, Issue 2–3, pp 269–294 | Cite as

A Mixed λ-calculus

Article
  • 39 Downloads

Abstract

The aim of this paper is to define a λ-calculus typed in aMixed (commutative and non-commutative) Intuitionistic Linear Logic. The terms of such a calculus are the labelling of proofs of a linear intuitionistic mixed natural deduction NILL, which is based on the non-commutative linear multiplicative sequent calculus MNL [RuetAbrusci 99]. This linear λ-calculus involves three linear arrows: two directional arrows and a nondirectional one (the usual linear arrow). Moreover, the -terms are provided with seriesparallel orders on free variables.

We prove a normalization theorem which explicitly gives the behaviour of the order during the normalization procedure.

Keywords

Typed λ-calculus non-commutative linear logic order varieties series-parallel orders normalization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bechet, de Groote, Rétoré 1997]
    Bechet, Denis, Philippe de Groote, and Christian Rétoré, A complete axiomatisation for the inclusion of series-parallel partial orders, in RTA’97 series LNCS 1232, 1997, pp. 230–240.Google Scholar
  2. [Buszkowski 1987]
    Buszkowski, W., ‘The Logic of Types’, in J. Srzednicki (ed.), Initiatives in Logic, M. Nijhoff, Dordrecht, 1987, pp. 180–206.Google Scholar
  3. [Buszkowski 1988]
    Buszkowski, W., ‘Generative Power of Categorial Grammars’, in R. Oehrle et al. (eds.), Categorial Grammars and Natural Languages Structures, Reidel, Dortrecht, 1988, pp. 69–94.Google Scholar
  4. [de Groote 2001]
    de Groote, Philippe, ‘Towards Abstract Categorial Grammars’, Association for Computational Linguistics, 39th Annual Meeting and 10th Conference of the European Chapter, Proceeding of the conference, pp. 148–155.Google Scholar
  5. [de Groote 96]
    de Groote, Philippe ‘Partially commutative linear logic: sequent calculus and phase semantics’, in Michele Abrusci and Claudia Casadio (eds.), Third Roma Workshop: Proofs and Linguistic Categories Applications of Logic to the analysis and implementation of Natural Language, Bologna: CLUEB, 1996, pp. 199–208.Google Scholar
  6. [Lambek 58]
    Lambek Joachim (1958). ‘The mathematics of sentences structures’. Am. Math. Monthly 65:154–169CrossRefGoogle Scholar
  7. [Mogbil 2001]
    Mogbil, Virgile, ‘Quadratic correctness criterion for non-commutative logic’, Lecture Notes in Computer Sciences 2142, 2001 proceeding of the 15th International Workshop Computer Science Logic, CSL 2001, 10th Annual Conference of the EACSL, Paris, France, September 10-13, 2001, pp. 69–83.Google Scholar
  8. [Mohring 89]
    Mohring, R., Computationally tractable classes of ordered sets, ASI Series 222, NATO, 1989.Google Scholar
  9. [Moortgat 97]
    Moortgat, Michael, ‘Categorial type logic’, in van Benthem and ter Meulen (eds.), Handbook of logic and language, North-Holland Elsevier, Amsterdam, Chapter 2, 1997, pp. 93–177.Google Scholar
  10. [Muskens 2001]
    Muskens, Reinart, ‘λ-grammars and the Syntax-Semantics Interface’, in Robert van Rooy and Martin Stokhof (eds.), Proceedings of the Thirteenth Amsterdam Colloquium, University of Amsterdam, 2001, pp. 150–155.Google Scholar
  11. [RuetAbrusci 99]
    Abrusci Vito Michele, Paul Ruet (1999). ‘Non-commutative logic I: the multiplicative fragment’. Annals of Pure and Applied Logic 101(1):29–64CrossRefGoogle Scholar
  12. [Ruet 2000]
    Ruet, Paul, ‘Non-commutative logic II: sequent calculus and phase semantics’, Math. Struct. in Comp. Sciences vol.10, Cambridge University Press, 2000, pp. 277–312.Google Scholar
  13. [Wansing 92]
    Wansing, Heinrich, ‘Formulas-as-Types for a hierarchy of sublogics of intuitionistic propositional logics’, in D. Pearce and H. Wansing (eds.), Non classical Logics and Information Processing, Springer Lecture Notes in AI 619:125–145, 1992, Springer, Berlin.Google Scholar
  14. [Wansing 93]
    Wansing Heinrich (1993). ‘The Logic of Information Structures’. Springer Lecture Notes in AI 681:59–80Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.IML-CNRS, Aix-Marseille UniversitéMarseille cedex 9France

Personalised recommendations