Space-Efficient Deterministic Simulation of Probabilistic Automata

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 the input string by that automaton? The question is interesting even in the case of one-head one-way probabilistic finite automata (PFA). We call (rational) stochastic languages (S^>_{rat}) the class of languages recognized by PFAs whose transition probabilities and cutpoints (i.e., recognition thresholds) are rational numbers. The class S^>_{rat} contains context-sensitive languages that are not context free, but on the other hand there are context-free languages not included in S^>_{rat}. Our main results are as follows: (1) The (proper) inclusion of S^>_{rat} in Dspace(log n), which is optimal (i.e., S^>_{rat} \not \subset Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972; Wang 1992] and Dspace(log n log log n) [Jung 1984]. (2) Probabilistic Turing machines with space bound f(n) \in O(log n) can be deterministically simulated in space O(min (c^{f(n)}log n, log n (f(n) + log log n))), where c is a constant depending on the simulated probabilistic Turing machine. The best previously known simulation required space O(log n (f(n) + log log n)) [Jung 1984]. Of independent interest is our technique to compare numbers given in terms of their values modulo a sequence of primes, p_1 < p_2 < \cdots < p_n = O(n^a) (where a is some constant) in O(log n) deterministic space