# Modified realizability toposes and strong normalization proofs

## Abstract

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* TOP_{m}(*U*) of non-standard sets equipped with an equality predicate. Church's Thesis is internally valid in *TOP*_{ m }(*K*_{1}) (where the *pca k*_{1} 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 **TOP**_{m}(*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.

## Keywords

Type Theory Strong Normalization Completeness Property Quotient Structure Realizability Category## Preview

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## References

- [B85]J. Bénabou. Fibred categories and the foundations of naïve category theory.
*J. Symb. Logic*, pages 10–37, 1985.Google Scholar - [Bar84]H. Barendregt.
*The Lambda Calculus*. North-Holland, revised edition, 1984.Google Scholar - [Bar91]H. Barendregt. Introduction to generalized type systems.
*J. Functional Prog.*, 1:125–154, 1991.Google Scholar - [Bee85]M. J. Beeson.
*Foundations of Constructive Mathematics*. Springer-Verlag, 1985.Google Scholar - [Bet87]I. Bethke. On the existence of extensional partial combinatory algebras.
*J. Symb. Logic*, pages 819–833, 1987.Google Scholar - [CH88]T. Coquand and G. Huet. The Calculus of Constructions.
*Info. and Comp.*, 76:95–120, 1988.Google Scholar - [Gal90]J. Gallier. On Girard's “Candidats de Réductibilité”. In P. Odifreddi, editor,
*Logic and Computer Science*. Academic Press, 1990.Google Scholar - [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
- [Gir86]J.-Y. Girard. The system
*F*of variable types, fifteen years later.*Theoretical Computer Science*, 45:159–192, 1986.Google Scholar - [Gra81]R. Grayson. Modified realizability toposes. unpublished manuscipt, 1981.Google Scholar
- [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 - [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 - [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 - [Hyl82]J. M. E. Hyland. The effective topos. In
*The L. E. J. Brouwer Centenary Symposium*, pages 165–216. North-Holland, 1982.Google Scholar - [Hyl88]J. M. E. Hyland. A small complete category.
*Annals of Pure and Applied Logic*, 40:135–165, 1988.Google Scholar - [Joh77]P. T. Johnstone.
*Topos Theory*. Academic Press, 1977. L.M.S. Monograph No. 10.Google Scholar - [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 - [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 - [ML73]P. Martin-Löf. An intuitionistic theory of types: Predicative part. In Rose and Shepherdson, editors,
*Logic Colloquium '73*. North-Holland, 1973.Google Scholar - [ML84]P. Martin-Löf.
*Intuitionistic Type Theory*. Bibliopolis, 1984. Studies in Proof Theory Series.Google Scholar - [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 - [Pra65]D. Prawitz.
*Natural Deduction*. Almqvist and Wiksell, 1965. Stockholm Studies in Philosophy 3.Google Scholar - [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 - [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 - [See87]R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus.
*J. Symb. Logic*, 52:969–989, 1987.Google Scholar - [Str92]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
- [Tai67]W. W. Tait. Intensional interpretation of functionals of finite type i.
*J. Symb. Logic*, 32:198–212, 1967.Google Scholar - [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 - [Tro73]A. Troelstra.
*Metatnathematical Investigation of Intuitionistic Arithmetic and Analysis*. Springer, 1973. Springer Lecture Notes in Mathematics 344.Google Scholar - [vO91]J. van Oosten.
*Exercises in Realizability*. PhD thesis, University of Amsterdam, 1991.Google Scholar