Abstract
Several authors have independently introduced second order theories whose provably total functionals are polynomial time computable functions on strings (e.g. [4], [6] and [7]), including the first author ([3], meant to be the second part of [2]). In this paper we give a detailed proof of the bi-interpretability result between such a second order theory and Buss' first order bounded arithmetic, based on an elegant definition of multiplication due to the second author.
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