Synchronization of finite automata

The thesis is an introduction to the research of sychronizing words of finite automata and the Černý conjecture. We give an overview of significant results in the field and their proof methods and describe a wide spectrum of unsolved problems. We reproduce also some latest results, namely Trakhtman's upper bound of general synchronizing threshold and Steinberg's tight bound for one-cluster automata with prime length cycle. In the last chapter we focus on subset synchronization and related computational problems. We give a new lower bound of maximal subset synchronizing threshold and partially determine the time complexity of a natural restriction of the PSPACE-complete problem deciding about subset synchronizability