  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.
