The Synchronization Problem for Strongly Transitive Automata

The synchronization problem for a deterministic n-state automaton consists in the search of an input-sequence, called a synchronizing word such that the state attained by the automaton, when this sequence is read, does not depend on the initial state of the automaton itself. If such a sequence exists, the automaton is called synchronizing. If the automaton is deterministic and complete, a well-known conjecture by Cerny claims that it has a synchronizing word of length not larger than (n− 1) [1]. This conjecture has been shown to be true for several classes of automata. Two results deserve a special mention: in [3], Kari proved Cerny conjecture for automata whose underlying graph is Eulerian. Dubuc [2] proved the conjecture for circular automata, that is, for automata possessing a letter that acts, as a circular permutation, over the set of states of the automaton. Recently, Béal proposed an unified algebraic approach that allows one to obtain quadratic bounds for the minimal length of a synchronizing word of circular or Eulerian automata. By developing this theoretical approach, we study the synchronization problem for a new class of automata called strongly transitive. A n-state automaton is said to be strongly transitive if it is equipped by a set of words {w1, ..., wn}, called independent, such that, for any pair of states s and t, there exists a word wi such that swi = t. An extension to unambiguous automata is also considered.