Modified realizability toposes and strong normalization proofs

Extended abstract
  • J. M. E. Hyland
  • C. -H. L. Ong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 664)


This paper is motivated by the discovery that an appropriate quotient SN of the strongly normalising untyped λ*-terms (where * is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinalory algebras (c-pca). Remarkably, an arbitrary rightabsorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizabilily, as opposed to the standard Kleene-style realizability. Starting from an arbitrary right-absorptive C-PCA U, the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm(U) of non-standard sets equipped with an equality predicate. Church's Thesis is internally valid in TOP m (K1) (where the pca k1 is “Kleene's first model” of natural numbers) but not Markov's Principle. There is a topos inclusion of SET-the “classical” topos of sets-into TOPm(U); the image of the inclusion is just sheaves for the ⌝⌝-topology. Separated objects of the ⌝⌝-topology are characterized. We identify the appropriate notion of PER's (partial equivalence relations) in the modified realizability setting and state its completeness properties. The topos TOP m (U) has enough completeness property to provide a category-theoretic semantics for a family of higher type theories which include Girard's System F and the Calculus of Constructions due to Coquand and Huet. As an important application, by interpreting type theories in the topos TOP m (SN.), a clean semantic explanation of the Tait-Girard style strong normalization argument is obtained. We illustrate how a strong normalization proof for an impredicative and dependent type theory may be assembled from two general “stripping arguments” in the framework of the topos TOP m (SN.). This opens up the possibility of a “generic” strong normalization argument for an interesting class of type theories.


Type Theory Strong Normalization Completeness Property Quotient Structure Realizability Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [B85]
    J. Bénabou. Fibred categories and the foundations of naïve category theory. J. Symb. Logic, pages 10–37, 1985.Google Scholar
  2. [Bar84]
    H. Barendregt. The Lambda Calculus. North-Holland, revised edition, 1984.Google Scholar
  3. [Bar91]
    H. Barendregt. Introduction to generalized type systems. J. Functional Prog., 1:125–154, 1991.Google Scholar
  4. [Bee85]
    M. J. Beeson. Foundations of Constructive Mathematics. Springer-Verlag, 1985.Google Scholar
  5. [Bet87]
    I. Bethke. On the existence of extensional partial combinatory algebras. J. Symb. Logic, pages 819–833, 1987.Google Scholar
  6. [CH88]
    T. Coquand and G. Huet. The Calculus of Constructions. Info. and Comp., 76:95–120, 1988.Google Scholar
  7. [Gal90]
    J. Gallier. On Girard's “Candidats de Réductibilité”. In P. Odifreddi, editor, Logic and Computer Science. Academic Press, 1990.Google Scholar
  8. [Gir72]
    J.-Y. Girard. Interprétation fonctionelle et elimination des coupures dans l'arithmétique d'order supérieur. Thèse de Doctorat d'Etat, Paris, 1972.Google Scholar
  9. [Gir86]
    J.-Y. Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159–192, 1986.Google Scholar
  10. [Gra81]
    R. Grayson. Modified realizability toposes. unpublished manuscipt, 1981.Google Scholar
  11. [HJP80]
    J. M. E. Hyland, P. T. Johnstone, and A. M. Pitts. Tripos theory. Math. Proc. Camb. Phil. Soc., 88:205–232, 1980.Google Scholar
  12. [HP89]
    J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. Contemporary Mathematics, 92:137–199, 1989.Google Scholar
  13. [HRR90]
    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.Google Scholar
  14. [Hyl82]
    J. M. E. Hyland. The effective topos. In The L. E. J. Brouwer Centenary Symposium, pages 165–216. North-Holland, 1982.Google Scholar
  15. [Hyl88]
    J. M. E. Hyland. A small complete category. Annals of Pure and Applied Logic, 40:135–165, 1988.Google Scholar
  16. [Joh77]
    P. T. Johnstone. Topos Theory. Academic Press, 1977. L.M.S. Monograph No. 10.Google Scholar
  17. [Kre59]
    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
  18. [LS86]
    J. Lambek and P. J. Scott. Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics No. 7. Cambridge University Press, 1986.Google Scholar
  19. [ML73]
    P. Martin-Löf. An intuitionistic theory of types: Predicative part. In Rose and Shepherdson, editors, Logic Colloquium '73. North-Holland, 1973.Google Scholar
  20. [ML84]
    P. Martin-Löf. Intuitionistic Type Theory. Bibliopolis, 1984. Studies in Proof Theory Series.Google Scholar
  21. [Pit87]
    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
  22. [Pra65]
    D. Prawitz. Natural Deduction. Almqvist and Wiksell, 1965. Stockholm Studies in Philosophy 3.Google Scholar
  23. [Rey74]
    J. C. Reynolds. Towards a theory of type structure. In B. Robinet, editor, Colloque sur la Programmation, pages 405–425. Springer-Verlag, 1974. Lecture Notes in Computer Science Vol. 19.Google Scholar
  24. [Sce89]
    A. Scedrov. Normalization revisited. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 357–369. AMS, 1989.Google Scholar
  25. [See87]
    R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Logic, 52:969–989, 1987.Google Scholar
  26. [Str92]
    T. Streicher. Truly intensional models of type theory arising from modified realizability. dated 25 May '92 mailing list at, 1992.Google Scholar
  27. [Tai67]
    W. W. Tait. Intensional interpretation of functionals of finite type i. J. Symb. Logic, 32:198–212, 1967.Google Scholar
  28. [Tai75]
    W. W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium. Springer-Verlag, 1975. Lecture Notes in Mathematics Vol. 453.Google Scholar
  29. [Tro73]
    A. Troelstra. Metatnathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, 1973. Springer Lecture Notes in Mathematics 344.Google Scholar
  30. [vO91]
    J. van Oosten. Exercises in Realizability. PhD thesis, University of Amsterdam, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • J. M. E. Hyland
    • 1
    • 3
  • C. -H. L. Ong
    • 2
    • 3
  1. 1.Dept. of Pure Maths. and Mathematical StatisticsCambridgeUK
  2. 2.Computer LaboratoryCambridgeUK
  3. 3.University of CambridgeEngland

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