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A generic strong normalization argument: Application to the Calculus of Constructions

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Computer Science Logic (CSL 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 832))

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Abstract

Hyland and Ong [HO93] recently showed that given an arbitrary (right-absorptive) conditionally partial combinatory algebra (Cpca), there is a systematic way to construct a Kreisel-style modified realizability topos which has more than sufficient completeness properties to provide a categorical semantics for a wide range of higher type theories. Based on the topos generated from the C-pca of an appropriate quotient of the strongly normalising untyped λ*-terms (where * is just a formal constant), they obtained a “generic” strong normalization argument. This argument reduces the strong normalization property of a class of higher type theories (up to F ω) to the validity of two “stripping arguments” which are reasonably easy to verify. This paper reports work which carries the same programme a step further. We illustrate the general applicability and simplicity of the argument by bringing it to bear on the demanding test case of Coquand and Huets' Calculus of Constructions.

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References

  1. J. Bénabou. Fibred categories and the foundations of naïve category theory. J. Symb. Logic, pages 10–37, 1985.

    Google Scholar 

  2. H. Barendregt. Introduction to generalized type systems. J. Functional Prog., 1:125–154, 1991.

    MATH  MathSciNet  Google Scholar 

  3. K. Petersson B. Nordström and J. M. Smith. Programming in Martin-Löf Type Theory: An Introduction. Oxford University Press, 1990.

    Google Scholar 

  4. J. Cartmell. Generalized algebraic theories and contextual categories. Annals of Pure and Applied Logic, 32, 1986.

    Google Scholar 

  5. A. Carboni, P. J. Freyd, and A. Scedrov. A categorical approach to realizability and polymorphic types. In M. Main et al., editor, Mathematical Foundations of Programming Language Semantics. Springer, 1988. LNCS Vol. 298.

    Google Scholar 

  6. T. Coquand and G. Huet. The Calculus of Constructions. Info. and Comp., 76:95–120, 1988.

    MathSciNet  Google Scholar 

  7. Th. Ehrhard. A categorical semantics of constructions. In Ieee Annual Symp. Logic in Computer Science. Computer Society Press, 1988.

    Google Scholar 

  8. Y. Ershov. Theorie der numerierungen I & II. Z. Math. Logik., 19 & 21, 1973 & 1975.

    Google Scholar 

  9. J. Gallier. Realizability, covers, and sheaves. unpublished manuscript, available by anonymous ftp, 1993.

    Google Scholar 

  10. H. Geuvers. The church-rosser property for βη-reduction in typed lambda calculi. In Proc. 7th Annual Ieee Symp. Logic in Computer Science, pages 453–460. Computer Societyy Press, 1992.

    Google Scholar 

  11. H. Geuvers. Logics and Type Systems. PhD thesis, University of Nijmegen, 1993.

    Google Scholar 

  12. J.-Y. Girard. Interprétation fonctionelle et elimination des coupures dans l'arithmetique d'order supérieur. Thèse de Doctorat d'Etat, Paris, 1972.

    Google Scholar 

  13. J.-Y. Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159–192, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  14. J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1989. Cambridge Tracts in Theoretical Computer Science 7.

    Google Scholar 

  15. J. M. E. Hyland, P. T. Johnstone, and A. M. Pitts. Tripos theory. Math. Proc. Camb. Phil. Soc., 88:205–232, 1980.

    MathSciNet  Google Scholar 

  16. J. M. E. Hyland and C.-H. L. Ong. Modified realizability topos and strong normalization proofs. In Proc. Conf. Type Lambda Calculus and Applications, Utrecht, March 93, pages 179–194. Springer-Verlag, 1993. LNCS Vol. 664.

    Google Scholar 

  17. J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. Contemporary Mathematics, 92:137–199, 1989.

    MathSciNet  Google Scholar 

  18. J. M. E. Hyland, E. P. Robinson, and G. Rosolini. The discrete objects in the effective topos. Proc. London Math. Soc. (3), 60:1–36, 1990.

    MathSciNet  Google Scholar 

  19. J. M. E. Hyland. A small complete category. Annals of Pure and Applied Logic, 40:135–165, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Jacobs. Categorical Type Theory. PhD thesis, Catholic University of Nijmegen, 1991.

    Google Scholar 

  21. G. Kreisel. Interpretation of analysis by means of constructive functionals of finite type. In A Heyting, editor, Constructivity in Mathematics. North-Holland, 1959.

    Google Scholar 

  22. G. Longo and E. Moggi. Constructive natural deducation and its “ω-set” interpretation. Math. Structures in Comp. Sc., ?, 1992.

    Google Scholar 

  23. Z. Luo. A higher-order calculus and theory abstraction. Information and Computation, 90:107–137, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. Milner. A theory of type polymorphism in programming. J. Computer and System Science, 17:348–375, 1978.

    MATH  MathSciNet  Google Scholar 

  25. P. Martin-Löf. About models for intuitionistic type theories and the notion of definitional equality, paper read at the Orléans Logic Conference, 1972.

    Google Scholar 

  26. P. Martin-Löf. Intuitionistic Type Theory. Bibliopolis, 1984. Studies in Proof Theory Series.

    Google Scholar 

  27. W. Phoa. An introduction to fibration, topos theory and the effective topos and modest sets. Technical Report ECS-LFCS-92-208, Edinburgh University, 1992.

    Google Scholar 

  28. A. M. Pitts. Polymorphism is set-theoretical, constructively. In D. H. Pitt et al., editor, Proc. Conf. Category Theory and Computer Science, Edinburgh, Berlin, 1987. Springer-Verlag. LNCS. Vol. 287.

    Google Scholar 

  29. A. M. Pitts. Categorical semantics of dependent types. Talk given at Sri Menlo Park, 2 June, 1989.

    Google Scholar 

  30. R. A. G. Seely. Locally cartesian closed categories and type theory. Math. Proc. Camb. Phil. Soc., 95, 1984.

    Google Scholar 

  31. R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Logic, 52:969–989, 1987.

    MATH  MathSciNet  Google Scholar 

  32. T. Streicher. Semantics of Type Theory: Correctness, Completeness and Independence Results. Birkhäuser, 1991.

    Google Scholar 

  33. T. Streicher. Truly intensional models of type theory arising from modified realizability, dated 25 May '92 mailing list at CATEGORIES@mta.ca, 1992.

    Google Scholar 

  34. W. W. Tait. Intensional interpretation of functionals of finite type i. J. Symb. Logic, 32:198–212, 1967.

    MATH  MathSciNet  Google Scholar 

  35. P. Taylor. Recursive Domains, Indexed Catgory Theory and Polymorphism. PhD thesis, Dpmms, University of Cambridge, 1987.

    Google Scholar 

  36. J. Terlouw. Strong normalization in type systems: a model-theoretic approach. In Dirk van Dalen Festschrift, pages 161–190. Dept. of Philosophy, Utrecht Univ., 1993.

    Google Scholar 

  37. S. Thompson. Type Theory and Functional Programming. Addison-Wesley, 1991.

    Google Scholar 

  38. A. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, 1973. Springer Lecture Notes in Mathematics 344.

    Google Scholar 

  39. R. Turner. Constructive Foundations for Functional Languages. McGraw Hill, 1991.

    Google Scholar 

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Authors and Affiliations

  1. Computing Laboratory, Oxford University, Wolfson Building, Parks Road, OX1 3QD, Oxford, England

    C. -H. L. Ong & E. Ritter

Authors
  1. C. -H. L. Ong
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  2. E. Ritter
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Editor information

Egon Börger Yuri Gurevich Karl Meinke

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© 1994 Springer-Verlag Berlin Heidelberg

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Ong, C.H.L., Ritter, E. (1994). A generic strong normalization argument: Application to the Calculus of Constructions. In: Börger, E., Gurevich, Y., Meinke, K. (eds) Computer Science Logic. CSL 1993. Lecture Notes in Computer Science, vol 832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0049336

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  • DOI: https://doi.org/10.1007/BFb0049336

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  • Print ISBN: 978-3-540-58277-9

  • Online ISBN: 978-3-540-48599-5

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