An Arithmetic Theory for the Poly-Time Random Functions

Melissa Antonelli; Ugo Dal Lago; Davide Davoli; Isabel Oitavem; Paolo Pistone

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arXiv · 2301.12028

An Arithmetic Theory for the Poly-Time Random Functions

Melissa Antonelli·Ugo Dal Lago·Davide Davoli·Isabel Oitavem·Paolo Pistone

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Abstract

We introduce a new bounded theory RS^1_2 and show that the functions which are Sigma^b_1-representable in it are precisely random functions which can be computed in polynomial time. Concretely, we pass through a class of oracle functions over string, called POR, together with the theory of arithmetic RS^1_2. Then, we show that functions computed by poly-time PTMs are arithmetically characterized by a class of probabilistic bounded formulas.

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Melissa Antonelli, Ugo Dal Lago, Davide Davoli, Isabel Oitavem, Paolo Pistone. 2023-01-27. An Arithmetic Theory for the Poly-Time Random Functions. https://arxiv.org/abs/2301.12028

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