The polynomial hierarchy and intuitionistic Bounded Arithmetic

  • Samuel R. Buss
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 223)


Intuitionistic theories IS 2 i of Bounded Arithmetic are introduced and it is shown that the definable functions of IS 2 i are precisely the □ i p functions of the polynomial hierarchy. This is an extension of earlier work on the classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast to the classical theories of Bounded Arithmetic where Σ i b -definable functions are of interest, our results for intuitionistic theories concern all the definable functions.

The method of proof uses □ i p -realizability which is inspired by the recursive realizability of S.C. Kleene [3] and D. Nelson [5]. It also involves polynomial hierarchy functionals of finite type which are introduced in this paper.


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  1. [1]
    S.R. Buss, Bounded Arithmetic, Ph.D. dissertation, Princeton University, 1985.Google Scholar
  2. [2]
    S.R. Buss, "The polynomial hierarchy and fragments of Bounded Arithmetic", 17th Annual ACM Symp. on Theory of Computing, Providence, R.I., pp. 285–290.Google Scholar
  3. [3]
    S.C. Kleene, "On the interpretation of intuitionistic number theory", Journal of Symbolic Logic, 10(1945), 109–124.Google Scholar
  4. [4]
    J.C.C. McKinsey, "Proof of the independence of the primitive symbols of Heyting's calculus of propositions", Journal of Symbolic Logic 4(1939), 155–158.Google Scholar
  5. [5]
    D. Nelson, "Recursive functions and intuitionistic number theory", Transactions of the American Mathematical Society, 61(1947), 307–368.Google Scholar
  6. [6]
    G. Takeuti, Proof Theory, North-Holland, 1975.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Samuel R. Buss
    • 1
  1. 1.Mathematical Sciences Research InstituteUSA

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