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25th Annual Symposium onFoundations of Computer Science, 1984. (1984)
Singer Island, FL
Oct. 24, 1984 to Oct. 26, 1984
ISBN: 0-8186-0591-X
pp: 323-331
A.G. Greenberg , AT&T Bell Laboratories
Freivalds recently reported a construction of a 2-way probabilistic finite automaton M that recognizes the set {a/sup m/b /sup m/ : m /spl ges/ 1} with arbitrarily small probability of error. This result implies that probabilistic machines of this type are more powerful than their deterministic, nondeterministic, and alternating counterparts. Freivalds' construction has a negative feature: the automation M runs in /spl Omega/ (2/sup n/2/n) expected time in the worst case on inputs of length n. We show that it is impossible to do significantly better. Specifically, no 2-way probabilistic finite automaton that runs in n/sup O (1)/ expected time recognizes {a/sup m/b/sup m/ : m /spl ges/ 1} with probability of error bounded away from 1/2. In passing we derive results on the densities of regular sets, the fine structure of Freivalds' construction, and the behavior of random walks controlled by Markov chains.

A. Greenberg and A. Weiss, "A Lower Bound For Probabilistic Algorithms For Finite State Machines," 25th Annual Symposium onFoundations of Computer Science, 1984.(FOCS), Singer Island, FL, 1984, pp. 323-331.
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