Space-efficient deterministic simulation of probabilistic automata

Abstract

Given a description of a probabilistic automaton (one-head probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of an input string by that automaton? The question is interesting even in the case of one-head one-way probabilistic finite automata. We call (rational) stochastic languages (Formula Found) the class of languages recognized by these devices with rational transition probabilities and rational cutpoint. Our main results are as follows: The (proper) inclusion of (Formula Found) in Dspace (log n), which is optimal (i.e. (Formula Found) Dspace (o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972], [Wang 1992] and Dspace(log n log log n) [Jung 1984]. The inclusion of the languages recognized by S(n) ϵ O(log n) spacebounded probabilistic Turing machines in Dspace(min(2S(n) log n, log n(S(n)+ loglogn))). The previous upper bound was Dspace(logn(S(n)+log logn)) [Jung 1984]. Of independent interest is our technique to compare numbers given in terms of their values modulo a sequence of primes, p1 < p2 < pn = O(na) (where a is some constant) in O(log n) deterministic space.