A Lower Bound for Probabilistic Algorithms for Finite State Machines

Freivalds recently reported a construction of a 2-way probabilistic finite automaton A4 that recognizes the set Iambm : m b 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 automaton A4 runs in O(2" ' 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 2way probabilistic finite automaton that runs in no(1) expected time recognizes (ambm : m b 11 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.