Space-Efficient Deterministic Simulation of Probabilistic Automata

• Authors: Ioan I. Macarie
• Venue: SIAM Journal on Computing
• Year: 1998

Given a description of a probabilistic automaton (one-head probabilistic finite-state 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-state automata (1pfa's). We call (rational) stochastic languages} ($\SS_{rat}^{}$) the class of languages recognized by 1pfa's whose transition probabilities and cutpoints (i.e., recognition thresholds) are rational numbers. The class $\SS_{rat}^{}$ contains context-sensitive languages that are not context-free, but on the other hand there are context-free languages not included in $\SS_{rat}^{}$.Our main results are as follows: \begin{itemize} \item{ The (proper) inclusion of $\SS_{rat}^{}$ in $\zlo$, which is optimal (i.e., $\SS_{rat}^{} \not \subset$ \Dspace$(o(\log n))$). The previous upper bounds were \Dspace$(n)$ (obtained by P. D. Dieu in 1972) and\linebreak[4]\Dspace$(\log n \log \log n)$ (obtained by H. Jung in 1984).} \item{ 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 uses O(log n (f(n) + log log n)) space (obtained by H. Jung in 1984).} \end{itemize}To obtain these results we develop and use two space-efficient (but also parallel-time-efficient) techniques, which are of independent interest: \begin{itemize} \item {a technique to compare deterministically, using only O(log n) space, large numbers given in terms of their values modulo a sequence of primes, p1 p2 pn in O(na) (where a is some arbitrary constant);} \item {a technique to compare deterministically a threshold with entries of the inverse of a given banded matrix.} \end{itemize}