The RAM equivalent of P vs. RP

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.Comment: 23 page

A correspondence between the time and space complexity

We investigate the correspondence between the time and space recognition complexity of languages; for this purpose, we will code the long-continued computations of deterministic two-tape Turing machines by the relatively short-length quantified Boolean formulae. The modified Stockmeyer and Meyer method will appreciably be used for this simulation. It will be proved using this modeling that the complexity classes $\mathbf{EXP}$ and $\mathbf{PSPACE}$ coincide; and more generally, the class $(k\!+\!1)$-fold Deterministic Exponential Time equals to the class $k$-fold Deterministic Exponential Space for each $k\geqslant1$; the space complexity of the languages of the class $\mathbf{P}$ will also be studied. Furthermore, this allows us to slightly improve the early founded lower complexity bound of decidable theories that are nontrivial relative to some equivalence relation (this relation may be equality) -- each of these theories is consistent with the formula, which asserts that there are two non-equivalent elements. Keywords: computational complexity, the coding of computations through formulae, exponential time, polynomial space, lower complexity bound of the language recognitionComment: 44 pages, 26 references bibliography; text overlap with arXiv:1907.04521 because the paper is created in the same metho