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Applied Categorical Structures

, Volume 13, Issue 1, pp 1–36 | Cite as

Relating Categorical Semantics for Intuitionistic Linear Logic

  • Maria Emilia Maietti
  • Paola Maneggia
  • Valeria de Paiva
  • Eike Ritter
Article

Abstract

There are several kinds of linear typed calculus in the literature, some with their associated notion of categorical model. Our aim in this paper is to systematise the relationship between three of these linear typed calculi and their models. We point out that mere soundness and completeness of a linear typed calculus with respect to a class of categorical models are not sufficient to identify the most appropriate class uniquely. We recommend instead to use the notion of internal language when relating a typed calculus to a class of models. After clarifying the internal languages of the categories of models in the literature we relate these models via reflections and coreflections.

Keywords

intuitionistic linear logic typed lambda calculus symmetric monoidal closed categories symmetric monoidal adjunctions 

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References

  1. 1.
    Barber, A.: Linear type theories, semantics and action calculi, Ph.D. thesis, University of Edinburgh, 1997. Google Scholar
  2. 2.
    Barber, A. and Plotkin, G.: Dual intuitionistic linear logic, Technical Report, LFCS, University of Edinburgh, 1997. Google Scholar
  3. 3.
    Barendregt, H. P.: Lambda calculi with types, in Handbook of Logic in Computer Science, Oxford Sci. Publ. 2, Oxford Univ. Press, New York, 1992, pp. 117–309. Google Scholar
  4. 4.
    Barr, M. and Wells, C.: Category Theory for Computing Science, Series in Computer Science, Prentice-Hall International, New York, 1990. Google Scholar
  5. 5.
    Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models, in Proceedings of Computer Science Logic ‘94 (Kazimierz, Poland), Lecture Notes in Comput. Sci. 933, Springer, Berlin, 1995. Google Scholar
  6. 6.
    Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models, Technical Report 352, University of Cambridge-Computer Laboratory, October 1994. Google Scholar
  7. 7.
    Benton, N., Bierman, G., de Paiva, V. and Hyland, M.: Linear λ-calculus and categorical models revisited, in Computer Science Logic (San Miniato, 1992), Lecture Notes in Comput. Sci. 702, 1993, pp. 75–90. Google Scholar
  8. 8.
    Benton, N., Bierman, G., de Paiva, V. and Hyland, M.: A term calculus for intuitionistic linear logic, in Proc. of Typed Lambda Calculus and Applications, Lecture Notes in Comput. Sci. 664, Springer, 1993, pp. 75–90. Google Scholar
  9. 9.
    Bierman, G.: On intuitionistic linear logic, Technical Report 346, Computer Laboratory, University of Cambridge, August 1994. Ph.D. thesis. Google Scholar
  10. 10.
    Bierman, G.: What is a categorical model of intuitionistic linear logic? in Proc. of the Second International Conference on Typed Lambda Calculus and Applications, Lecture Notes in Comput. Sci. 902, Springer, 1994. Google Scholar
  11. 11.
    Crole, R. L.: Categories for Types, Cambridge Mathematical Textbooks, Cambridge University Press, 1993. Google Scholar
  12. 12.
    Ghani, N.: Adjoint rewriting, Ph.D. thesis, University of Edinburgh, 1995. Published as CST-122-95 and ECS-LFCS-95-339. Google Scholar
  13. 13.
    Ghani, N., de Paiva, V. and Ritter, E.: Linear explicit substitutions, J. IGPL (2000). Google Scholar
  14. 14.
    Girard, J. Y. and Lafont, Y.: Linear logic and lazy computation, in TAPSOFT ‘87, Lecture Notes in Comput. Sci. 250, 1987, pp. 52–66. Google Scholar
  15. 15.
    Girard, J. Y., Lafont, Y. and Taylor, P.: Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, 1989. Google Scholar
  16. 16.
    Hyland, M. and Schalk, A.: Abstract games for linear logic, Extended Abstract, in CTCS ‘99, Electronic Notes in Theoret. Comput. Sci. 29, 1999. Google Scholar
  17. 17.
    Kelly, G. M.: Doctrinal adjunctions, in Category Seminar ‘73, Lecture Notes in Math. 420, 1974, pp. 257–281. Google Scholar
  18. 18.
    Kelly, G. M.: Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Note Ser. 64, Cambridge University Press, 1982. Google Scholar
  19. 19.
    Koh, T. W. and Ong, C.-H. L.: Explicit substitution internal languages for autonomous and *-autonomous categories (preliminary version), in Proceedings of the 8th Conference on Category Theory and Computer Science, Electronic Notes in Theoretical Computer Science 29, 1999, 30 pp. Google Scholar
  20. 20.
    Lafont, Y.: The linear abstract machine, in International Joint Conference on Theory and Practice of Software Development (Pisa, 1987), Theoret. Comput. Sci. 59, 1988, pp. 157–180. Google Scholar
  21. 21.
    Lambek, J. and Scott, P. J.: An Introduction to Higher Order Categorical Logic, Studies in Adv. Math. 7, Cambridge University Press, 1986. Google Scholar
  22. 22.
    Mac Lane, S.: Categories for the Working Mathematician, Graduate Text in Math. 5, Springer, 1971. Google Scholar
  23. 23.
    Mackie, I., Roman, L. and Abramsky, S.: An internal language for autonomous categories, Appl. Categorical Structures 1 (1993), 311–343. Google Scholar
  24. 24.
    Maietti, M. E., de Paiva, V. and Ritter, E.: Categorical models for intuitionistic and linear type theory, in J. Tiuryn (ed.), Proc. of Foundations of Software Science and Computation Structures, Lecture Notes in Comput. Sci., 2000. Google Scholar
  25. 25.
    Maneggia, P.: Models of linear polymorphism, Ph.D. thesis, The University of Birmingham, UK, 2004. Google Scholar
  26. 26.
    Mellies, P. A.: Categorical models of linear logic revisited, Theoret. Comput. Sci. (2002) (submitted). Google Scholar
  27. 27.
    Pitts, A. M.: Categorical logic, in Logical Methods in Computer Science, Handbook of Logic in Computer Science, Vol. VI, Oxford University Press, 1995. Google Scholar
  28. 28.
    Ritter, E. and de Paiva, V.: Short final summary of EPSRC-grant: The eXplicit linear abstract machine, University of Birmingham, http://www.cs.bham.ac.uk/research/xslam/ xslam_summary.ps.gz, June 2000.
  29. 29.
    Street, R.: The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149–168. Google Scholar
  30. 30.
    Taylor, P.: Practical Foundations of Mathematics, Cambridge Stud. in Adv. Math. 99, Cambridge University Press, 1997. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Maria Emilia Maietti
    • 1
  • Paola Maneggia
    • 2
  • Valeria de Paiva
    • 3
  • Eike Ritter
    • 2
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di PadovaItaly
  2. 2.School of Computer ScienceUniversity of BirminghamUK
  3. 3.Palo Alto Research CenterUSA

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