Abstract
We consider the following generalized notion of synchroniz ation: A word is called a reset word of a subset of states of a deterministic finite automaton if it ma ps all states of the set to a unique state. It is known that the minimum length of such words is superpoly nomial in worst cases, namely in a series of substantially nontransitive automata.
We prese nt a series of transitive binary automata with a strongly exponential minimum length. This also const itutes a progress in the research of composition sequences initiated by Arto Salomaa, because r eset words of subsets are just a special case of composition sequences.
Deciding about the existenc e of a reset word for given automaton and subset is known to be a PSPACE-complete problem, we prove that this holds even if we restrict the problem to transitive binary automata.