Structure in Complexity Theory pp 77-103 | Cite as

# The polynomial hierarchy and intuitionistic Bounded Arithmetic

## Abstract

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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## References

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*Bounded Arithmetic*, Ph.D. dissertation, Princeton University, 1985.Google Scholar - [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
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