# A syntax for linear logic

Conference paper

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## Abstract

There is a standard syntax for Girard's linear logic, due to Abramsky, and a standard semantics, due to Seely. Alas, the former is incoherent with the latter: different derivations of the same syntax may be assigned different semantics. This paper reviews the standard syntax and semantics, and discusses the problem that arises and a standard approach to its solution. A new solution is proposed, based on ideas taken from Girard's Logic of Unity. The new syntax is based on pattern matching, allowing for concise expression of programs.

## Keywords

Pattern Match Linear Logic Natural Deduction Linear Assumption Concrete Syntax
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## References

- [Abr90]S. Abramsky, Computational interpretations of linear logic. Presented at
*Workshop on Mathematical Foundations of Programming Language Semantics*, 1990. To appear in*Theoretical Computer Science*.Google Scholar - [AJ92]S. Abramsky and R. Jagadeesan, New foundations for the geometry of interaction. In
*7'th Symposium on Logic in Computer Science*, IEEE Press, Santa Cruz, California, June 1992.Google Scholar - [Bar79]M. Barr, *-
*Autonomous Categories*. Lecture Notes in Mathematics 752, Springer Verlag, 1979.Google Scholar - [BBdPH92]N. Benton, G. Bierman, V. de Paiva, and M. Hyland, Type assignment for intuitionistic linear logic. Draft paper, August 1992.Google Scholar
- [BCGS91]V. Breazu-Tannen, T. Coquand, C. A. Gunter, and A. Scedrov, Inheritance as explicit coercion.
*Information and Compututation*, 93:172–221, 1991. (An earlier version appeared in*Symposium on Logic in Computer Science*, IEEE Press, Asilomar, California, June 1989.)Google Scholar - [BTKP93]V. Breazu-Tannen, D. Kesner, L. Puel, A typed pattern calculus. In
*8'th Symposium on Logic in Computer Science*, Montreal, June 1993.Google Scholar - [CGR92]J. Chirimar, C. A. Gunter, and J. G. Riecke. Linear ML. In
*Symposium on Lisp and Functional Programming*, ACM Press, San Francisco, June 1992.Google Scholar - [Fil92]A. Filinski, Linear continuations. In
*Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, New Mexico, January 1992.Google Scholar - [Gir87]
- [Gir91]J.-Y. Girard, On the unity of logic. Manuscript, 1991.Google Scholar
- [Hol88]S. Holmström, A linear functional language. Draft paper, Chalmers University of Technology, 1988.Google Scholar
- [Laf88]Y. Lafont, The linear abstract machine.
*Theoretical Computer Science*,**59**:157–180, 1988.Google Scholar - [LM92]P. Lincoln and J. Mitchell, Operational aspects of linear lambda calculus. In
*7'th Symposium on Logic in Computer Science*, IEEE Press, Santa Cruz, California, June 1992.Google Scholar - [LS91]Y. Lafont and T. Streicher. Game semantics for linear logic. In
*6'th Symposium on Logic in Computer Science*, IEEE Press, Amsterdam, July 1991.Google Scholar - [Mac91]I. Mackie, Lilac: a functional programming language based on linear logic. Master's Thesis, Imperial College London, 1991.Google Scholar
- [Mog89]E. Moggi, Computational lambda-calculus and monads. In
*4'th Symposium on Logic in Computer Science*, IEEE Press, Asilomar, California, June 1989.Google Scholar - [O'He91]P. W. O'Hearn, Linear logic and interference control. In
*Conference on Category Theory and Computer Science*, Paris, September 1991. LNCS, Springer Verlag.Google Scholar - [Pra91]V. Pratt, Event spaces and their linear logic. In
*AMAST '91: Algebraic Methodology And Software Technology*, Iowa City, Springer Verlag LNCS, 1992.Google Scholar - [Red91]U. Reddy, Acceptors as Values. Manuscript, December 1991.Google Scholar
- [See89]R. A. G. Seely, Linear logic, *-autonomous categories, and cofree coalgebras. In
*Categories in Computer Science and Logic*, June 1989. AMS Contemporary Mathematics 92.Google Scholar - [Tro92]A. S. Troelstra,
*Lectures on Linear Logic*. CSLI Lecture Notes, 1992.Google Scholar - [Wad90]P. Wadler, Linear types can change the world! In M. Broy and C. Jones, editors,
*Programming Concepts and Methods*, Sea of Galilee, Israel, North Holland, April 1990.Google Scholar - [Wad91]P. Wadler, Is there a use for linear logic? In
*Conference on Partial Evaluation and Semantics-Based Program Manipulation (PEPM)*, New Haven, Connecticut, ACM Press, June 1991.Google Scholar - [Wad92]P. Wadler, There's no substitute for linear logic. Presented at
*Workshop on Mathematical Foundations of Programming Language Semantics*, Oxford, April 1992.Google Scholar - [Wad93]P. Wadler, A taste of linear logic. In
*Mathematical Foundations of Computer Science*, Gdansk, Poland, LNCS, Springer Verlag, August 1993.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1994