P-RAM vs. RP-RAM

Abstract

One of the fundamental open questions in computational complexity is whether the class of problems solvable by use of stochasticity under the Random Polynomial time (RP) model is larger than the class of those solvable in deterministic polynomial time (P). However, this question is only open for Turing machines, not for Random Access Machines (RAMs). Simon (1981) [31] was able to show that for a sufficiently equipped Random Access Machine, the ability to switch states nondeterministically entails no computational advantage. On the other hand, in the same paper, Simon describes a different (and arguably more natural) model for RAM stochasticity. According to Simon's proposal, instead of receiving a new random bit at each execution step, the RAM program is able to execute the pseudofunction RAND(y), which returns a uniformly distributed random integer in the range [0,y). Whether the ability to allot a random integer in this fashion is more powerful than the ability to allot a random bit remained an open question for the last 30 years.In this paper, we close Simon's open problem by fully characterising the class of languages recognisable in polynomial time by each of the RAMs regarding which the question was posed. We show that for some of these stochasticity does not entail any advantage, but, more interestingly, we show that for others it does. These results carry over also to BPP-like and coRP-like acceptance criteria.