A logical aspect of parametric polymorphism

  • Ryu Hasegawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1092)


The system of formal parametric polymorphism has the same theory as second order Peano arithmetic with regard to the provable equality of numerical functions.


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  1. 1.
    M. Abadi, L. Cardelli, and P.-L. Curien, Formal parametric polymorphism, Theoret. Comput. Sci. 121 (1993) 9–58.Google Scholar
  2. 2.
    E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott, Functorial polymorphism, Theoret. Comput. Sci. 70 (1990) 35–64; Corrigendum, 71 (1990) 431.Google Scholar
  3. 3.
    H. P. Barendregt, The lambda calculus, Its syntax and semantics, revised edition, (North-Holland, 1984).Google Scholar
  4. 4.
    C. Böhm and A. Berarducci, Automatic synthesis of typed Λ-programs on term algebras, Theoret. Comput. Sci. 39 (1985) 135–154.Google Scholar
  5. 5.
    P. Freyd, Structural polymorphism, Theoret. Comput. Sci. 115 (1993) 107–129.Google Scholar
  6. 6.
    J.-Y. Girard, Interprétation Fonctionnelle et Élimination des Coupures de l'Arithmétique d'Ordre Supérieur, Thèse d'Etat, Université Paris VII (1972).Google Scholar
  7. 7.
    J.-Y. Girard, Proof Theory and Logical Complexity, Volume I, Studies in Proof Theory, (Bibliopolis, 1987).Google Scholar
  8. 8.
    J.-Y. Girard, P. Taylor and Y. Lafont, Proofs and Types, (Cambridge University Press, 1989).Google Scholar
  9. 9.
    K. Gödel, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12 (1958) 280–287; English translation in J. Philosophical Logic 9 (1980) 133–142; reprinted in Kurt Gödel, Collected Works, Volume II, Publications 1938–1974, S. Feferman et al., eds., (Oxford University Press, 1990) pp. 240–251.Google Scholar
  10. 10.
    R. Hasegawa, Categorical data types in parametric polymorphism, Math. Struct. Comput. Sci. 4 (1994) 71–109.Google Scholar
  11. 11.
    R. Hasegawa, Relational limits in general polymorphism, Publ. Research Institute for Mathematical Sciences 30 (1994) 535–576.Google Scholar
  12. 12.
    J. R. Hindley and J. P. Seldin, Introduction to Combinators and λ-Calculus, (London Math. Soc., 1986).Google Scholar
  13. 13.
    S. C. Kleene, Introduction to Metamathematics, (North-Holland, 1964).Google Scholar
  14. 14.
    D. Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, in: IEEE 24th Annual Symp. on Foundations of Computer Science, (IEEE, 1983) pp. 460–469.Google Scholar
  15. 15.
    R. Loader, Models of linear logic and inductive datatypes, Mathematical Institute, Oxford University, preprint (1994).Google Scholar
  16. 16.
    G. Plotkin and M. Abadi, A logic for parametric polymorphism, in: Typed Lambda Calculi and Applications, M. Bezem, J. F. Groote, eds., 1993, Utrecht, The Netherlands, Lecture Notes in Computer Science 664, (Springer, 1993) 361–375.Google Scholar
  17. 17.
    J. C. Reynolds, Towards a theory of type structure, in: B. Robinet ed., Programming Symposium, Paris, Lecture Notes in Computer Science 19 (Springer, 1974) pp. 408–425.Google Scholar
  18. 18.
    J. C. Reynolds, Types, abstraction, and parametric polymorphism, in: Information Processing 83, R. E. A. Mason, ed., (North-Holland, 1983) pp. 513–523.Google Scholar
  19. 19.
    R. A. G. Seely, Categorical semantics for higher order polymorphic lambda calculus, J. Symbolic Logic 52 (1987) 969–989.Google Scholar
  20. 20.
    R. M. Smullyan, Theory of Formal Systems, Annals of Mathematics Studies 47, (Princeton University Press, 1961).Google Scholar
  21. 21.
    C. Spector, Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, in: Recursive Function Theory, J. C. E. Dekker, ed., Proceedings of Symposia in Pure Mathematics, Volume V, (AMS, 1962) pp. 1–27.Google Scholar
  22. 22.
    R. Statman, Number theoretic functions computable by polymorphic programs (extended abstract), in: IEEE 22nd Annual Symp. on Foundations of Computer Science, Los Angels, 1981 (IEEE, 1981) pp. 279–282.Google Scholar
  23. 23.
    A. S. Troelstra, Mathematical Investigations of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics 344, (Springer, 1973).Google Scholar
  24. 24.
    A. S. Troelstra, Introductory note to 1958 and 1972, in: Kurt Gödel, Collected Works, Volume II, Publications 1938–1974, S. Feferman et al., eds., (Oxford University Press, 1990) pp. 217–241.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Ryu Hasegawa
    • 1
  1. 1.Oxford University Computing LaboratoryOxfordEngland

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