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On Proof Terms and Embeddings of Classical Substructural Logics

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Abstract

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of the right structural rules. In terms of Parigot's λμ-calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the λμ-terms into four categories. It is proved that well-typed GLx-λμ-terms correspond to GLx proofs, and that a GLx-λμ-term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Gödel-style translations of GLx-λμ-terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx-λμ-terms are strongly normalizable.

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References

  1. H. P. Barendregt, ‘Lambda calculi with types’, Handbook of Logic in Computer Science, Oxford University Press, 1991.

  2. M. W. Bunder, ‘Theorems in classical logic are instances of theorems in Condensed BCI Logic’, in Substructural Logics, edited by P. Schroeder-Heister and K. Do{ie219-1}en, Clarendon Press, p. 43–62, 1993.

  3. K. Do{ie219-2}en, ‘A historical introduction to substructural logics’, in Substructural Logics, edited by P. Schroeder-Heister and K. Do{ie219-3}en, Clarendon Press, p. 259–291, 1993.

  4. M. Dummett, Elements of Intuitionism, Clarendon Press, 1977.

  5. K. Fujita, On Embedding of Classical Substructural Logics, Rewriting theory and its applications, RIMS no. 918, Kyoto University, p. 178–195, 1995.

  6. K. Fujita, ‘Strong normalization of pure GL w-λμ-terms’, A Study of Non-Classical Logic Based on Sequent Calculus and Kripke Semantics, RIMS no. 1021, Kyoto University, f. 1–13, 1997.

  7. H-Y. Girard, ‘Linear logic’, Theoretical Computer Science 50, 1–102, 1987.

    Google Scholar 

  8. T. G. Griffin, ‘A formulae-as-types notion of control’, in Proc. the 17th ACM Symposium on Principles of Programming Languages, p. 47–58, 1990.

  9. P. de Groote, ‘A CPS-translation of the λμ-calculus’, Lecture Notes in Computer Science 787, 85–99, 1994.

  10. J. R. Hindley, The Principal-Type-Scheme Algorithm Revisited: Principal Deductions in λ-Calculus, Mathematics Division, University College, Swansea, 20 July, 1988.

    Google Scholar 

  11. J. R. Hindley, ‘BCK-combinators and linear λ-terms have types’, Theoretical Computer Science 64, 97–105, 1989.

    Google Scholar 

  12. J. R. Hindley and D. Meredith, ‘Principal type-schemes and condensed detachment’, The Journal of Symbolic Logic vol. 55,no. 1, 90–105, 1990.

    Google Scholar 

  13. J. R. Hindley Basic Simple Type Theory, Cambridge University Press, 1997.

  14. W. Howard, ‘The formulae-as-types notion of construction’, to H. B. Curry, Essays on Combinatory Logic, Lambda-Calculus, and Formalizm, Academic Press, p. 479–490, 1980.

  15. S. Kuroda, ‘Intuitionistische untersucheungen der formalistischen Logik’, Nagoya Mathematical Journal vol. 2, 35–47, 1951.

    Google Scholar 

  16. C. R. Murthy, ‘An evaluation semantics for classical proofs’, Proc. of 6th IEEE Symposium on Logic in Computer Science, p. 96–107, 1991.

  17. C.-H. L. Ong, A Semantic View of Classical Proofs: Type-Theoretic, Categorical, and Denotational Characterizations, Linear Logic '96 Tokyo Meeting, Keio University, 1996.

  18. H. Ono, ‘Structural rules and a logical hierarchy’, Proc. of Heyting '88 Conference, Plenum Press, p. 95–104, 1990.

  19. H. Ono, ‘Semantics of substructural logics’, in Substructural Logics, edited by P. Schroeder-Heister and K. Došen, Clarendon Press, p. 259–291, 1993.

  20. M. Parigot, ‘λμ-calculus: an algorithmic interpretation of classical natural deduction’, Lecture Notes in Computer Science 624, 190–201, 1992.

  21. M. Parigot, ‘Strong normalisation for second order classical natural deduction’, Proc. the 8th IEEE symposium on Logic in Computer Science, 39–46, 1993.

  22. M. Parigot, ‘Classical proofs as programs’, Lecture Notes in Computer Science 713, 263–276, 1993.

  23. D. Prawitz, Natural Deduction: A Proof-Theoretical Study, Almqvist & Wiksell, Stockholm 1965.

    Google Scholar 

  24. N. J. Rehof and M. H. SØrensen, ‘The λΔ-calculus’ Lecture Notes in Computer Science 789, 516–542, 1994.

  25. G. Gentzen, The Collected Papers of Gerhard Gentzen, edited by M. E. Szabo, North-Holland, 1969.

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  1. Department of Artificial Intelligence, Kyushu Institute of Technology, Iizuka, 820-8502, Japan

    Ken-etsu Fujita

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  1. Ken-etsu Fujita
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Fujita, Ke. On Proof Terms and Embeddings of Classical Substructural Logics. Studia Logica 61, 199–221 (1998). https://doi.org/10.1023/A:1005073330585

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