On the power of Las Vegas for one-way communication complexity, OBDDs, and finite automata.

*(English)*Zbl 1007.68065Summary: The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.

(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) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.

(iii) 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 square root of the size of the minimal deterministic finite automaton recognizing \(L\). Using a specific language we verify the optimality of this lower bound.

(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) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.

(iii) 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 square root of the size of the minimal deterministic finite automaton recognizing \(L\). Using a specific language we verify the optimality of this lower bound.

##### MSC:

68Q10 | Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) |

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

68W20 | Randomized algorithms |

##### Keywords:

computational power of randomized computations; Las Vegas computation; deterministic computation; nondeterministic computation; complexity measures; one-way communication complexity; ordered binary decision diagrams; finite automata
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\textit{J. Hromkovič} and \textit{G. Schnitger}, Inf. Comput. 169, No. 2, 284--296 (2001; Zbl 1007.68065)

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