On the power of Las Vegas II: Two-way finite automata

Abstract The investigation of the computational power of randomized computations is one of the central tasks of complexity and algorithm theory. While for one-way finite automata the power of different computational modes was successfully determined, one does not have any nontrivial result relating the power of determinism, Las Vegas and nondeterminism for two-way finite automata. The main results of this paper are as follows. (i) If, for a regular language L , there exist small two-way nondeterministic finite automata for both L and L ʗ , then there exists a small two-way Las Vegas finite automaton for L . (ii) There is a quadratic gap between nondeterminism and Las Vegas for two-way finite automata. (iii) For every k ∈ N , there is a regular language S k such that S k can be accepted by a two-way Las Vegas finite automaton with O (k) states, but every two-way deterministic finite automaton recognizing S k has at least Ω (k 2 / log 2 k) states.