Running time to recognize nonregular languages by 2-way probabilistic automata

Abstract

R. Freivalds proved that the language {0^m1^m} can be recognized by 2-way probabilistic finite automata (2pfa) with arbitrarily high probability 1-ɛ. A.G.Greenberg and A.Weiss proved that no 2pfa can recognize this language in expected time $T(n) = c^\circ{(n)}$. For arbitrary languages C.Dwork and L.Stockmeyer showed somewhat less: if a language L is recognized by a 2pfa in expected time $T(n) = c^{n^\circ{(1)} }$, then L is regular. First, we improve this theorem replacing the expected time by the time with probability 1-ɛ. On the other hand, time bound by C.Dwork and L.Stockmeyer cannot be improved: for arbitrary k≥2 we exhibit a specific nonregular language that can be recognized by 2pfa with probability 1-ɛ in expected time $T(n) = c^{n^{{1 \mathord{\left/{\vphantom {1 k}} \right.\kern-\nulldelimiterspace} k}} }$. Finally, we show that the running time for the recognition of nonregular 2-dimensional languages by 4-way pfa can be essentially smaller, namely, linear.