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.

翻译：我们引入了一种新的有界理论RS^1_2，并证明其中可Sigma^b_1表示的函数恰好是可以在多项式时间内计算的随机函数。具体而言，我们通过一类基于字符串的预言机函数类POR，结合算术理论RS^1_2，证明了由多项式时间概率图灵机计算的函数可被一类概率有界公式算术地刻画。