A categorical approach to realizability and polymorphic types

  • Aurelio Carboni
  • Peter J. Freyd
  • Andre Scedrov
Part I Categorical And Algebraic Methods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)


A categorical calculus of relations is used to derive a unified setting for higher order logic and polymorphic lambda calculus.


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  1. Bainbridge E. S., Freyd P. J., Scedrov A., Scott P. J. [88], Functorial polymorphism. In: "Logical Foundations of Functional Programming" (G. Huet, ed.), Addison-Wesley, to appear.Google Scholar
  2. Beeson M. [1985], "Foundations of constructive mathematics". Ergeb. der Math., Springer-Verlag, New York.Google Scholar
  3. Boileau A., Joyal A. [1981], La logique des topos. J. Symb. Logic 46, 6–16.Google Scholar
  4. Breazu-Tannen V., Coquand T. [87], Extensional models for polymorphism. Proc. TAPSOFT '87 — CFLP, Pisa, Theor. Comp. Sci., to appear.Google Scholar
  5. Bruce K. B., Meyer A. R., Mitchell J. C. [87], The semantics of second-order lambda calculus. Information & Computation, to appear.Google Scholar
  6. Carboni A., Walters R. F. C. [88], Cartesian bicategories I. J. Pure Appl. Algebra, to appear.Google Scholar
  7. Cardelli L., Wegner P. [1985], On understanding types, data abstraction, and polymorphism. ACM Comp. Surveys 17 (4) 471–522.CrossRefGoogle Scholar
  8. Cousineau G., Curien P. L., Robinet B. (eds.) [86], "Combinators and functional programming languages". Springer LNCS 242.Google Scholar
  9. Freyd P. J. [74], On functorializing model theory, or: on canonizing category theory. Manuscript.Google Scholar
  10. Freyd P. J., Kelly G. M. [1972], Categories of continuous functors I. J. Pure Appl. Alg. 2, 169–191; Erratum, ibid. 4 (1974), 121.CrossRefGoogle Scholar
  11. Freyd P. J., Scedrov, A. [1987], Some semantic aspects of polymorphic lambda calculus. Proc. 2nd IEEE Symp. Logic in Computer Science, Ithaca, N.Y., pp. 315–319.Google Scholar
  12. Freyd P.J., Scedrov A. [1988], "Geometric logic". Math. Library Ser., North-Holland, Amsterdam, to appear.Google Scholar
  13. Girard J.Y. [1971], Une extension de l'interprétation de l'interprétation de Gődel ... Proc. 2nd Scandinavian Logic Symp. (J. Fenstad, ed.), North-Holland, Amsterdam, 63–92.Google Scholar
  14. Girard J.Y. [72], "Interprétation fonctionelle et élimination des coupures ...". Thèse de Doctorat d'Etat. Université Paris VII.Google Scholar
  15. Hyland J.M.E. [1982], The effective topos. Proc. Brouwer Centenary Symposium (A. S. Troelstra, D. van Dalen, eds.), North-Holland, Amsterdam, 165–216.Google Scholar
  16. Hyland J.M.E., Robinson, E., Rosolini, G. [87], Discrete objects in a topos. Preprint.Google Scholar
  17. Kleene S.C. [1959], Recursive functionals and quantifiers of finite type I. Trans. Amer. Math. Soc. 91, 1–52.Google Scholar
  18. Lambek J., Scott P. [86], "Introduction to higher-order categorical logic". Cambridge Univ. Press.Google Scholar
  19. McCarty D.C. [1986], Realizability and recursive set theory. Ann. Pure Appl. Logic 32, 153–183.CrossRefGoogle Scholar
  20. Mitchell J.C. [86], A type-inference approach to reduction properties and semantics of polymorphic expressions. Symp. on Lisp and Functional Programming.Google Scholar
  21. Reynolds J. C. [1974], Towards a theory of type structure. Springer LNCS 19, 408–425.Google Scholar
  22. Reynolds J.C. [84], Polymorphism is not set-theoretic. Springer LNCS 173.Google Scholar
  23. Rosolini G. [86], About modest sets. PreprintGoogle Scholar
  24. Scedrov A. [88], Recursive realizability and the calculus of constructions. In: "Logical Foundations of Functional Programming" (G. Huet, ed.), Addison-Wesley, to appear.Google Scholar
  25. Scott D.S. [82], Domains for denotational semantics. ICALP 82. Springer LNCS 140.Google Scholar
  26. Seely R.A.G. [87], Categorical semanticsfor higher-order polymorphic lambda calculus. J. Symb. Logic, to appear.Google Scholar
  27. Wagner E. [1969], Uniformly reflective structures: on the nature of gődelizations and relative computability. Trans. Amer. Math. Soc. 144, 1–41.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Aurelio Carboni
    • 1
  • Peter J. Freyd
    • 2
  • Andre Scedrov
    • 2
  1. 1.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
  2. 2.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaU.S.A.

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