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Chu’s Construction: A Proof-Theoretic Approach

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Logic for Concurrency and Synchronisation

Part of the book series: Trends in Logic ((TREN,volume 15))

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Abstract

The essential interaction between classical and intuitionistic features in the system of linear logic is best described in the language of category theory. Given a symmetric monoidal closed category C with products, the category C×Cop can be given the structure of a *-autonomous category by a special case of the Chu construction. The main result of the paper is to show that the intuitionistic translations induced by Girard’s trips determine the functor from the free *-autonomous category A on a set of atoms {P,P′,...} to C × Cop, where C is the free monoidal closed category with products and coproducts on the set of atoms {P O , P I , P′ O , P′ I ,...} (a pair P O , P I in C for each atom P of A).

Research supported by EPSRC senior research fellowship on grant GL/L 33382. We wish to thank Martin Hyland for his essential suggestions and crucial support in the develoment of this work.

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References

  1. Samson Abramsky and Radha Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic. J. Symb. Logic 59(2):543–574, 1994.

    Google Scholar 

  2. M. Barr. *-Autonomous categories, Lecture Notes in Mathematics 752, Springer-Verlag, 1979, Berlin, Heidelberg, New York.

    Google Scholar 

  3. M. Barr. *-Autonomous categories and linear logic, Mathematical Structures in Computer Science 1:159–178, 1991.

    Google Scholar 

  4. N. Benton, G. Bierman, V. de Paiva, M. Hyland. A term calculus for intuitionistic linear logic. Springer LNCS 664, pp. 75–90, 1993.

    Google Scholar 

  5. N. Benton, G. Bierman, V. de Paiva, M. Hyland. Linear λ-Calculus and Categorical Models Revisited, Preprint, Comp. Lab., Univ. of Cambridge.

    Google Scholar 

  6. G. M. Bierman. What is a Categorical Model of Intuitionistic Linear Logic? In Proceedings of the International Conference on Typed Lambda Calculi and Applications. April 10–12, 1995. Edinburgh, Scotland. Springer LNCS.

    Google Scholar 

  7. G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh.

    Google Scholar 

  8. G. Bellin and P. J. Scott. Theor. Comp. Sci. 135(1): 11–65, 1994.

    Article  Google Scholar 

  9. G. Bellin and J. van de Wiele. Subnets of Proof-nets in MLL, in Advances in Linear Logic, Girard, Lafont and Regnier eds., London Math. Soc. Lect. Note Series 222, Cambridge University Press, 1995, pp. 249–270.

    Google Scholar 

  10. V. Danos and L. Regnier. The Structure of Multiplicatives, Arch. Math. Logic 28:181–203, 1989.

    Article  Google Scholar 

  11. H. Devarajan, D. Hughes, G. Plotkin and V. Pratt. Full Completeness of the multiplicative linear logic of Chu spaces. Porceedings of LICS 1999.

    Google Scholar 

  12. Valeria C.V. de Paiva. The Dialectica Categories. PhD thesis. DPMMS, University of Cambridge, 1988. Available as Comp. Lab. Tech. Rep. 213, 1990.

    Google Scholar 

  13. A. Fleury. La règle d’échange. Thèse de doctorat, 1996, Équipe de Logique, Université de Paris 7, Paris, France.

    Google Scholar 

  14. A. Fleury and C. Retoré. The Mix Rule, Mathematical Structures in Computer Science 4:273–85, 1994.

    Google Scholar 

  15. H. N. Gabow and R. E. Tarjan. A Linear-Time Algorithm for a Special Case of Disjoint Set Union. Journal of Computer and System Science 30:209–221, 1985.

    Google Scholar 

  16. J-Y. Girard. Linear Logic, Theoretical Computer Science 50:1–102, 1987.

    Article  Google Scholar 

  17. J-Y. Girard. On the unity of logic. Ann. Pure App. Log. 59:201–217, 1993.

    Google Scholar 

  18. J-Y. Girard. Proof-nets: the parallel syntax for proof-theory. In Logic and Algebra, New York, 1995. Marcel Dekker.

    Google Scholar 

  19. M. Hyland and L. Ong. Fair games and full completeness for Multiplicative Linear Logic without the Mix rule. ftp-able at theory.doc.ic.ac.uk in papers/Ong, 1993.

    Google Scholar 

  20. F. Lamarche. Proof Nets for Intuitionistic Linear Logic 1: Essential Nets. Preprint ftp-able from Hypatia 1994.

    Google Scholar 

  21. F. Lamarche. Games Semantics for Full Propositional Logic. Proceedings of LICS 1995.

    Google Scholar 

  22. R. Loader. Models of Lambda Calculi and Linear Logic: Structural, Equational and Proof-Theoretic Characterisations, PhD Thesis, St. Hugh’s College, Oxford, UK, 1994.

    Google Scholar 

  23. A. S. Murawski and C.-H. L. Ong. A Linear-time Algorithm for Verifying MLL Proof Nets via Lamarche’s Essential Nets. Preprint, OUCL, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. andrzej@comlab.ox.ac.uk, http://www.comlab.ox.ac.uk/oucl/people/luke.ong.html, 1999.

    Google Scholar 

  24. A. Patterson. Implicit Programming and the Logic of Constructible Duality, PhD Thesis, University of Illinois at Urbana-Champaign, 1998. http://www.formal.stanford.edu/annap/www/abstracts.html#9

  25. V. Pratt. Chu spaces as a semantic bridge between linear logic and mathematics. Preprint ftp-able from http://boole.stanford.edu/chuguide.html, 1998.

  26. A. Tan. Full completeness for models of linear logic. PhD Thesis, King’s College, University of Cambridge, UK, October 1997.

    Google Scholar 

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© 2003 Kluwer Academic Publishers

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Bellin, G. (2003). Chu’s Construction: A Proof-Theoretic Approach. In: de Queiroz, R.J.G.B. (eds) Logic for Concurrency and Synchronisation. Trends in Logic, vol 15. Springer, Dordrecht. https://doi.org/10.1007/0-306-48088-3_2

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  • DOI: https://doi.org/10.1007/0-306-48088-3_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-1270-9

  • Online ISBN: 978-0-306-48088-1

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