Synchronizing Automata and the Černý Conjecture

Abstract

We provide a survey of research surrounding the Černý conjecture. This conjecture concerns finite-state automata that have the property of being "synchronizing." A synchronizing automaton is one for which there exists some input sequence that causes the automaton to transition to some fixed state, irrespective of the state in which it had been before reading that input sequence. The Černý conjecture states that, if an automaton with n states is synchronizing, then there exists some input sequence of length at most (n - 1)² that synchronizes it. We first survey the basic results that deal with synchronization of finite-state automata. We also study and implement several related algorithms, including Eppstein's greedy algorithm for producing a reset sequence. An analysis of the length of the sequence produced by this algorithm leads to an interesting problem in extremal combinatorics, the solution of which yields an upper bound of (n³ - n)/6 on the length of the sequence. We then investigate a generalization of the Černý conjecture. Next, we study extremal automata, for which the length of the shortest synchronizing sequence meets the Černý bound. We then turn our attention to subclasses of automata for which the Černý conjecture has been proved. Finally, we discuss possible approaches and current efforts to proving the Černý conjecture, as well as some related problems from the literature.