Lambda Calculus and Intuitionistic Linear Logic
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.
This paper introduces a typed functional language Λ! and a categorical model for it.
The terms of Λ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λ-abstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of Λ!, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language Λ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction.
The categorical model induces operational equivalences like, for example, a set of extensional equivalences.
The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the Principal-Type Property.
- Abramsky, S. (1990) Computational interpretation of linear logic. Department of Computing, Imperial College, London
- Abramsky, S., Jagadeesan, R. (1992) Games and full completeness for multiplicative linear logic. Department of Computing, Imperial College, London
- H.P. Barendregt, 1984, The Lambda Calculus, North-Holland, second edition.
- N. Benton, G. Bierman, V. de Paiva, and M. Hyland, 1990, Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August.
- V. Breazu-Tannen, D. Kesner, and L. Puel, 1993, A typed pattern calculus, In Proceedings of the 8th Symposium on Logic in Computer Science LICS'93 (Montreal), pages 262–274, June.
- J. Gallier, 1990, Logic and Computer Science, chapter On Girard's “candidats de reductibilités”, pages 123–203, P. Odifreddi editor, Academic Press.
- J. Gallier, 1993, On the correspondence between proofs and lambda terms, Obtained by ftp, January.
- J.Y. Girard, 1972, Interpretation Fonctionelle et Elimination des Coupures de l'Arithmetique d'Ordre Superieur, PhD thesis, Université Paris VII.
- Girard, J.Y. (1987) Linear logic. Theoretical Computer Science 50: pp. 1-102
- J.Y. Girard, Y. Lafont, and P. Taylor, 1989, Proofs and Types, Cambridge University Press.
- Huet, G. (1980) Confluent reductions: abstract properties and applications to term rewriting systems. Journal of A.C.M. 27: pp. 797-821
- B. Jacobs, 1992, Semantics of weakening and contraction, In Typed Lambda Calculi and Applications TLCA'92, volume LNGS. Springer-Verlag.
- Lafont, Y. (1988) The linear abstract machine. Theoretical Computer Science 59: pp. 157-180
- P. Lincoln and J. Mitchell, 1992, Operational aspects of linear lambda calculus, In Proceedings of Symposium on Logic in Computer Science LICS'92, pages 235–246, June.
- Simone Martini and Andrea Masini, 1993, On the fine structure of the exponential rule. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, pages 197–210. Cambridge University Press, 1995. Proceedings of the Workshop on Linear Logic, Ithaca, New York, June.
- G. Mints, Normal deductions in the intuitionistic linear logic. To appear in Archive for Mathematical Logic.
- A. Pravato and L. Roversi, 1995, A! considered both as a paradigmatic language and as a meta-language, In Theoretical Computer Science: Proceedings of the Fifth Italian Conference (Salerno), pages 146–161. World Scientific, November.
- Prawitz, D. (1965) Natural Deduction, a Proof Theoretic Study. Almquist and Wiksell, Amsterdam
- J.A. Reynolds, 1974, Paris Colloquium on Programming, chapter Towards a Theory of Type Structures, pages 408–425. Springer-Verlag.
- S. Ronchi della Rocca and L. Roversi 1994, Lambda calculus and intuitionistic linear logic, Invited talk at the Logic Colloquium'94 (Clermont-Ferrand), July.
- L. Roversi, 1996, Curry-Howard isomorphism and intuitionistic linear logic, Technical Report 19/96, Università degli Studi di Torino.
- A.S. Troelstra, 1992, Lectures on Linear Logic, CSLI.
- P. Wadler, 1993 A syntax for linear logic, Presented at the Mathematical Foundations of Programming Semantics, New Orleans, April.
- Lambda Calculus and Intuitionistic Linear Logic
Volume 59, Issue 3 , pp 417-448
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- linear logic
- Curry-Howard isomorphism
- typed λ-calculi