Exponential time and bounded arithmetic
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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