The Cerny Conjecture for Automata with Blocking States

In [7], Jan Černý conjectured that an arbitrary directable automaton with n states has a directing word of length not longer than (n−1). This conjecture is one of the most longstanding open problems in the theory of finite automata. Most of papers dealing with this conjecture reduce the problem to special classes of automata. In present paper we deal with this conjecture in the class of automata having a blocking state. We prove that the conjecture is true in this class of automata. We show that if an automaton has n states and contains a blocking state then it has a directing word whose length is not longer than n(n−1) 2 . The notion of the blocking state is a generalization of the notion of the trap for directable automata. Thus every trap-directable automaton with n states has a directing word of length not longer than n(n−1) 2 . We give an example for trap-directable automaton with n states in which the length of the shortest directing word is n(n−1) 2 . We prove that if the Černý Conjecture holds for a subautomaton of a directable automaton then it holds for the automaton.