Las Vegas versus determinism for one-way communication complexity, finite automata, and polynomial-time computations

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main aim of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. An at most polynomial gap has been established for the combinational complexity of circuits and for the communication complexity of two-party protocols. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, finite automata and polynomial-time relativized Turing machine computation. (i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight. (ii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound. Note, that this result establishes for the first time an at most polynomial gap between Las Vegas and determinism for a uniform computing model. (iii) For relativized polynomial computations we show that Las Vegas can be even more powerful than nondeterminism with a polynomial restriction on the number of nondeterministic guesses. On the other hand superlogarithmic many advice bits in nondeterministic computations can be more powerful than Las Vegas (even Monte Carlo) computations in a relativized word.