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Fast Synchronization of Random Automata

Author Cyril Nicaud

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Cyril Nicaud

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Cyril Nicaud. Fast Synchronization of Random Automata. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 43:1-43:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.43

Abstract

A synchronizing word for an automaton is a word that brings  that automaton into one and the same state, regardless of the starting position. Cerny conjectured in 1964 that if a $n$-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n-1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of  synchronization: he proved  that, for the uniform distribution on deterministic automata with n states,  an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^{3}n) with high probability. As a consequence, this proves that most automata satisfy the Cerny conjecture.

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Keywords
  • random automata
  • synchronization
  • the Černý conjecture

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