An Arithmetic Theory for the Poly-Time Random Functions

TL;DR

This paper introduces a new bounded arithmetic theory, RS^1_2, that characterizes polynomial-time computable random functions through a novel arithmetical framework involving probabilistic formulas.

Contribution

It establishes a precise correspondence between Sigma^b_1-representable functions in RS^1_2 and polynomial-time random functions, linking computational complexity with bounded arithmetic.

Findings

Characterization of polynomial-time random functions via RS^1_2

Introduction of POR oracle functions within the theory

Formal link between probabilistic polynomial-time Turing machines and arithmetic formulas

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.