Given a description of a probabilistic automaton (one-head probabilistic nite 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 nite automata (PFA). We call (rational) stochastic languages (S > rat ) the class of languages recognized by PFA's whose

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... on 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: