Polynomial Time with Restricted Use of Randomness

We define a hierarchy of complexity classes that lie between P and RP, yielding a new way of quantifying partial progress towards the derandomization of RP. A standard approach in derandomization is to reduce the number of random bits an algorithm uses. We instead focus on a model of computation that allows us to quantify the extent to which random bits are being used. More specifically, we consider Stack Machines (SMs), which are log-space Turing Machines that have access to an unbounded stack, an input tape of length N , and a random tape of length N. We parameterize these machines by allowing at most r(N)−1 reversals on the random tape, thus obtaining the r(N)-th level of our hierarchy, denoted by RPdL[r]. It follows by a result of Cook [Coo71] that RPdL[1] = P, and of Ruzzo [Ruz81] that RPdL[exp(N)] = RP. Our main results are the following. • For every i ≥ 1, derandomizing RPdL[2O(logi ] implies the derandomization of RNC. Thus, progress towards the P vs RP question along our hierarchy implies also progress towards derandomizing RNC. Perhaps more surprisingly, we also prove a partial converse: Pseurorandom generators (PRGs) for RNC are sufficient to derandomize RPdL[2 i ]; i.e. derandomizing using PRGs a class believed to be strictly inside P, we derandomize a class containing P. More generally, we introduce Randomness Compilers, a model equivalent to Stack Machines. In this model a polynomial time algorithm gets an input x and it outputs a circuit Cx, which takes random inputs. Acceptance of x is determined by the acceptance probability of Cx. When Cx is of polynomial size and depth O(logN) the corresponding class is denoted by P+RNC, and we show that RPdL[2 i ] ⊆ P+RNC ⊆ RPdL[2O(logi+1 ]. • We show an unconditional N lower bound on the number of reversals required by a SM for Polynomial Evaluation. This in particular implies that known Schwartz-Zippel-like algorithms for Polynomial Identity Testing cannot be implemented in the lowest levels of our hierarchy. • We show that in the 1-st level of our hierarchy, machines with one-sided error are as powerful as machines with two-sided and unbounded error.