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International Workshop on Types for Proofs and Programs

TYPES 1993: Types for Proofs and Programs pp 108–126Cite as

Logic of refinement types

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 806))

Abstract

Refinement types are subsets of ordinary types, which are intended to be specifications of programs. Ordinary types correspond to constructive propositions by Curry-Howard isomorphism. Refinement types correspond to “classical” propositions by a semantics resembling interpretations of logics in categorical/algebraic logic. In this paper, we will study the logic of refinement types in the type system ATTT which was introduced in [9] as a framework for an “optimized” Curry-Howard isomorphism.

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References

  1. M.J. Beeson, Foundations of Constructive Mathematics, Springer-Verlag, 1985.

    Google Scholar 

  2. R.M. Burstall and J. H. McKinna, Deliverables: an approach to program in Constructions, Technical Report ECS-LFCS-91-133, Department of Computer Science, The University of Edinburgh, 1991.

    Google Scholar 

  3. Dowek, C. et al., The Coq Proof Assistant User's Guide, Version 5.6, Technical Report No. 134, INRIA, December, 1991.

    Google Scholar 

  4. Th. Coquand and G. Huet, Calculus of Constructions, Information and Computation, vol. 76, pp. 95–120, 1988.

    Article  Google Scholar 

  5. T. Freeman and F. Pfenning, Refinement Types for ML, ACM SIGPLAN'91, Conference on Programming Language Design and Implementation, Toronto, Ontario, ACM Press, 1991.

    Google Scholar 

  6. S. Hayashi, On derived rules of intuitionistic second order arithmetic, Commentariorum Mathematicorum Universitatis Sancti Pauli, vol. XXVI, pp. 77–103, 1977.

    Google Scholar 

  7. S. Hayashi, Adjunction of semifunctors: categorical structures in non-extensional lambda calculus, Theoretical Computer Sciences, vol. 41, pp.95–104, 1986.

    Google Scholar 

  8. S. Hayashi and H. Nakano, PX: A Computational Logic, The MIT Press, 1988.

    Google Scholar 

  9. S. Hayashi, Singleton, Union and Intersection Types for Program Extraction, Lecture Notes in Computer Science No. 526, pp. 701–730, T.Ito and A.R.Meyer, eds., Springer-Verlag, 1991, an extended version to appear in Information and Computation.

    Google Scholar 

  10. R. Hoofman, The theory of semi-functors, Mathematical Structures in Computer Science, vol. 3, pp.93–128, 1993.

    Google Scholar 

  11. B. Jacobs, Semantics of the second order lambda calculus, Mathematical Structures in Computer Science, vol. 1, pp. 327–360, 1991.

    Google Scholar 

  12. Z. Luo. ECC, an Extended Calculus of Constructions, in Proceedings of the Fourth Annual IEEE Symposium on Logic in Computer Science, Asilomar, California, 1989.

    Google Scholar 

  13. Z. Luo. An Extended Calculus of Constructions, Ph. D. thesis, Department of Computer Science, The University of Edinburgh, 1990.

    Google Scholar 

  14. Z. Luo. A problem of adequacy: conservativity of Calculus of Constructions over higher order logic, Technical Report ECS-LFCS-90-121, Department of Computer Science, The University of Edinburgh, 1990.

    Google Scholar 

  15. M. Makkai and G. Reyes. First Order Categorical Logic, Lecture Note in Mathematics No. 611, 1977, Springer-Verlag.

    Google Scholar 

  16. J. H. McKinna. Deliverables: A Categorical Approach to Program Development in Type Theory, Ph. D. thesis, Department of Computer Science, The University of Edinburgh, 1992.

    Google Scholar 

  17. , H. Nakano. A Constructive Formalization of the Catch and Throw Mechanism, in Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, 1992.

    Google Scholar 

  18. C. Paulin-Mohring and B. Werner, Synthesis of ML programs in the System Coq, Journal of Symbolic Computation, 1993.

    Google Scholar 

  19. F. Pfenning, Intersection Types for a Logical Framework, in this volume, 1992.

    Google Scholar 

  20. Erik Poll, A programming logic for Fω, Computing Science Notes, 92/25, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1992.

    Google Scholar 

  21. A.S. Troelstra, Metamathematical investigations of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Ohtsu, 520-21, Shiga, Japan

    Susumu Hayashi

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  1. Susumu Hayashi
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Editor information

Henk Barendregt Tobias Nipkow

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To the memory of Prof. Ken Hirose

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© 1994 Springer-Verlag Berlin Heidelberg

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Hayashi, S. (1994). Logic of refinement types. In: Barendregt, H., Nipkow, T. (eds) Types for Proofs and Programs. TYPES 1993. Lecture Notes in Computer Science, vol 806. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58085-9_74

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  • DOI: https://doi.org/10.1007/3-540-58085-9_74

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58085-0

  • Online ISBN: 978-3-540-48440-0

  • eBook Packages: Springer Book Archive

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