What is a categorical model of Intuitionistic Linear Logic?

  • G. M. Bierman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 902)

Abstract

This paper re-addresses the old problem of providing a categorical model for Intuitionistic Linear Logic (ILL). In particular we compare the now standard model proposed by Seely to the lesser known one proposed by Benton, Bierman, Hyland and de Paiva. Surprisingly we find that Seely's model is unsound in that it does not preserve equality of proofs. We shall propose how to adapt Seely's definition so as to correct this problem and consider how this compares with the model due to Benton et al.

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References

  1. 1.
    S. Abramsky. Computational interpretations of linear logic. Theoretical Computer Science, 111(1–2):3–57, 1993. Previously Available as Department of Computing, Imperial College Technical Report 90/20, 1990.CrossRefGoogle Scholar
  2. 2.
    M. Barr. ⋆-autonomous categories and linear logic. Mathematical Structures in Computer Science, 1:159–178, 1991.Google Scholar
  3. 3.
    P.N. Benton. A mixed linear and non-linear logic: Proofs, terms and models. Technical Report 352, Computer Laboratory, University of Cambridge, 1994.Google Scholar
  4. 4.
    P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.Google Scholar
  5. 5.
    P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. A term calculus for intuitionistic linear logic. In M. Bezem and J.F. Groote, editors, Proceedings of Conference on Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 75–90, 1993.Google Scholar
  6. 6.
    G.M. Bierman. On Intuitionistic Linear Logic. PhD thesis, Computer Laboratory, University of Cambridge, December 1993. Available as Computer Laboratory Technical Report 346. August 1994.Google Scholar
  7. 7.
    J. Gallier. Constructive logics part I: A tutorial on proof systems and typed λ-calculi. Theoretical Computer Science, 110(2):249–339, March 1993.CrossRefGoogle Scholar
  8. 8.
    J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1–101, 1987.CrossRefGoogle Scholar
  9. 9.
    J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In Proceedings of TAPSOFT 87, volume 250 of Lecture Notes in Computer Science, pages 52–66, 1987. Previously Available as INRIA Report 588, 1986.Google Scholar
  10. 10.
    W.A. Howard. The formulae-as-types notion of construction. In J.R. Hindley and J.P. Seldin, editors, To H.B. Curry: Essays on combinatory logic, lambda calculus and formalism. Academic Press, 1980.Google Scholar
  11. 11.
    Y. Lafont. The linear abstract machine. Theoretical Computer Science, 59:157–180, 1988. Corrections ibid. 62:327–328, 1988.CrossRefGoogle Scholar
  12. 12.
    J. Lambek and P.J. Scott. Introduction to higher order categorical logic, volume 7 of Cambridge studies in advanced mathematics. Cambridge University Press, 1987.Google Scholar
  13. 13.
    S. Mac Lane. Categories for the Working Mathematican, volume 5 of Graduate Texts in Mathematics. Springer Verlag, 1971.Google Scholar
  14. 14.
    R.A.G. Seely. Linear logic, ⋇-autonomous categories and cofree algebras. In Conference on Categories in Computer Science and Logic, volume 92 of AMS Contemporary Mathematics, pages 371–382, June 1989.Google Scholar
  15. 15.
    A.S. Troelstra. Lectures on Linear Logic, volume 29 of Lecture Notes. CSLI, 1992.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • G. M. Bierman
    • 1
  1. 1.University of Cambridge Computer LaboratoryUK

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