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.
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- S. Abramsky, 1990, Computational interpretation of linear logic, Technical Report 90/92, Department of Computing, Imperial College, London.Google Scholar
- S. Abramsky and R. Jagadeesan, 1992, Games and full completeness for multiplicative linear logic, Technical Report 92/24, Department of Computing, Imperial College, London, September.Google Scholar
- H.P. Barendregt, 1984, The Lambda Calculus, North-Holland, second edition.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- J. Gallier, 1990, Logic and Computer Science, chapter On Girard's “candidats de reductibilités”, pages 123–203, P. Odifreddi editor, Academic Press.Google Scholar
- J. Gallier, 1993, On the correspondence between proofs and lambda terms, Obtained by ftp, January.Google Scholar
- J.Y. Girard, 1972, Interpretation Fonctionelle et Elimination des Coupures de l'Arithmetique d'Ordre Superieur, PhD thesis, Université Paris VII.Google Scholar
- J.Y. Girard, 1987, Linear logic Theoretical Computer Science, 50:1–102.Google Scholar
- J.Y. Girard, Y. Lafont, and P. Taylor, 1989, Proofs and Types, Cambridge University Press.Google Scholar
- G. Huet, 1980, Confluent reductions: abstract properties and applications to term rewriting systems, Journal of A.C.M., 27:797–821.Google Scholar
- B. Jacobs, 1992, Semantics of weakening and contraction, In Typed Lambda Calculi and Applications TLCA'92, volume LNGS. Springer-Verlag.Google Scholar
- Y. Lafont, 1988, The linear abstract machine, Theoretical Computer Science, 59:157–180.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- G. Mints, Normal deductions in the intuitionistic linear logic. To appear in Archive for Mathematical Logic.Google Scholar
- 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.Google Scholar
- D. Prawitz, 1965. Natural Deduction, a Proof Theoretic Study, Almquist and Wiksell-Amsterdam.Google Scholar
- J.A. Reynolds, 1974, Paris Colloquium on Programming, chapter Towards a Theory of Type Structures, pages 408–425. Springer-Verlag.Google Scholar
- S. Ronchi della Rocca and L. Roversi 1994, Lambda calculus and intuitionistic linear logic, Invited talk at the Logic Colloquium'94 (Clermont-Ferrand), July.Google Scholar
- L. Roversi, 1996, Curry-Howard isomorphism and intuitionistic linear logic, Technical Report 19/96, Università degli Studi di Torino.Google Scholar
- A.S. Troelstra, 1992, Lectures on Linear Logic, CSLI.Google Scholar
- P. Wadler, 1993 A syntax for linear logic, Presented at the Mathematical Foundations of Programming Semantics, New Orleans, April.Google Scholar