Synchronizing Automata: Open Problems

We survey selected open problems in the theory of synchronizing automata, centered around the famous Černý conjecture.A deterministic finite automaton is called synchronizing if it admits a reset word whose action maps all states to a single state.The Černý conjecture states that every synchronizing automaton with n states possesses a reset word of length at most (n -1) 2 .We discuss avoiding words, compressing a state with another, synchronization of a (given or any) subset, complexity of deciding the synchronizability, average reset threshold, and linear-algebraic methods.Some new auxiliary results are also presented.