Abstract
Intuitionistic theories IS i2 of Bounded Arithmetic are introduced and it is shown that the definable functions of IS i2 are precisely the □ pi 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 bi -definable functions are of interest, our results for intuitionistic theories concern all the definable functions.
The method of proof uses □ pi -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.
Research supported in part by NSF Grant DMS 85-11465.
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References
S.R. Buss, Bounded Arithmetic, Ph.D. dissertation, Princeton University, 1985.
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.
S.C. Kleene, "On the interpretation of intuitionistic number theory", Journal of Symbolic Logic, 10(1945), 109–124.
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.
D. Nelson, "Recursive functions and intuitionistic number theory", Transactions of the American Mathematical Society, 61(1947), 307–368.
G. Takeuti, Proof Theory, North-Holland, 1975.
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© 1986 Springer-Verlag Berlin Heidelberg
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Buss, S.R. (1986). The polynomial hierarchy and intuitionistic Bounded Arithmetic. In: Selman, A.L. (eds) Structure in Complexity Theory. Lecture Notes in Computer Science, vol 223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16486-3_91
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DOI: https://doi.org/10.1007/3-540-16486-3_91
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