A Lower Bound For Probabilistic Algorithms For Finite State Machines
Albert G. Greenberg, Alan A. Weiss · 2005
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.