The RAM equivalent of P vs. RP

TL;DR

This paper investigates the computational power of Random Access Machines (RAMs) with stochastic capabilities, specifically whether using a pseudorandom function like RAND(y) offers an advantage over deterministic RAMs, closing a 30-year open problem.

Contribution

The paper fully characterizes the classes of languages recognizable by RAMs with stochastic RAND(y) functions, determining when stochasticity provides computational advantages.

Findings

Stochastic RAMs can be more powerful than deterministic ones for certain classes.

For some RAM models, randomness does not increase computational power.

The paper resolves Simon's 1981 open problem regarding stochastic RAMs.

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) was able to show that for a sufficiently equipped Random Access Machine, the ability to switch states nondeterministically does not entail any computational advantage. However, in the same paper, Simon describes a different (and arguably more natural) scenario for stochasticity under the RAM model. 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 $\textit{RAND}(y)$, 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 entails no advantage, but, more interestingly, we show that for others it does.