Structure in Complexity Theory pp 125-143 | Cite as

# Exponential time and bounded arithmetic

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## Abstract

For each *n*, we give weak theories of bounded arithmetic, whose provably recursive functions (having appropriate graphs) are exactly those functions computable deterministically in *n*-fold time TIME(2(*n,p*(|*x*|))), where *p* is a polynomial and 2(*n,z*) is a stack of *n* two's topped by a *z*. In proving this result, we separate out the *time contribution* due to different variables in a multivariate function. These results further the evidence that “normalized” formal logic proofs (free cut free proof in Gentzen sequent calculus) of the totality of a function furnish an algorithm to compute the function.

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© Springer-Verlag Berlin Heidelberg 1986