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Comparing the theory of representations and constructive mathematics

  • A. S. Troelstra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 626)

Abstract

The paper explores the analogy between reducibility statements of Weihrauch's theory of representations and theorems of constructive mathematics which can be reformulated as inclusions between sets. Kleene's function-realizability is the key to understanding of the analogy, and suggests an alternative way of looking at the theory of reducibilities.

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References

  1. [B]
    M.J.Beeson, Foundations of Constructive Mathematics, Springer-Verlag, Berlin 1985.Google Scholar
  2. [BB]
    E. Bishop, D.S. Bridges, Constructive Analysis, Springer-Verlag, Berlin 1985.Google Scholar
  3. [Bi]
    E.A. Bishop, Foundations of Constructive Analysis (1967), McGraw Hill, New York.Google Scholar
  4. [Fr]
    H.M. Friedman, On the derivability of instantiation properties, The Journal of Symbolic Logic 42 (1977), 506–514.Google Scholar
  5. [K]
    S.C. Kleene, Formalized recursive functionals and formalized realizability, memoirs of the American mathematical Society 89 (1969).Google Scholar
  6. [KW1]
    C. Kreitz, K. Weihrauch, A unified approach to constructive and recursive analysis, in: M.M. Richter et al. (eds.), Computation and Proof theory, Springer-Verlag, Berlin 1984, 259–278.Google Scholar
  7. [KW2]
    C. Kreitz, K. Weihrauch, Theory of representations, Theoretical Computer Science 38 (1985), 35–53.Google Scholar
  8. [KW3]
    C. Kreitz, K. Weihrauch, Compactness in constructive analysis revisited, Annals of Pure and Applied Logic 36 (1987), 29–38.Google Scholar
  9. [M]
    N.Th. Müller, Computational complexity of real functions and real numbers. Informatik Berichte 59 (1986), Fern-Universität Hagen, BRD.Google Scholar
  10. [ML]
    P. Martin-Löf, Intuitionistic Type Theory, Bibliopolis, Napoli.Google Scholar
  11. [Sh]
    N.A. Shanin, Constructive Real Numbers and Function Spaces, American Mathematical Society, providence (RI), 1968 (translation of the russian original).Google Scholar
  12. [T]
    A.S. Troelstra (ed.), Metamathematical investigation of Intuitionistic Arithmetic and Analysis, Springer verlag, berlin 1973.Google Scholar
  13. [TD]
    A.S. Troelstra, D. van Dalen, Constructivism in Mathematics (1988), North-Holland Publ. Co., AmsterdamGoogle Scholar
  14. [vO]
    J. van Oosten, Exercises in Realizability. Ph.D. thesis, Universiteit van Amsterdam, 1990.Google Scholar
  15. [W1]
    K. Weihrauch, Type 2 recursion theory, Theoretical Computer Science 38 (1985), 17–33.Google Scholar
  16. [W2]
    K. Weihrauch, Computability on computable metric spaces, Theoretical Computer Science, to appear.Google Scholar
  17. [WK]
    K. Weihrauch, C. Kreitz, Representations of the real numbers and of the open subsets of the set of real numbers, Annals of Pure and Applied Logic 35 (1987), 247–260.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • A. S. Troelstra
    • 1
  1. 1.Faculteit Wiskunde en InformaticsUniversiteit van AmsterdamAmsterdamNL

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