Some Results Concerning Careful Synchronization of Partial Automata and Subset Synchronization of DFA’s

The goal of this paper is to present a family of partial automata that achieve length $$\varTheta (3^{\frac{n}{3}})$$ of the shortest carefully synchronizing words, but using $$\frac{2}{9}n + 2$$ letters, thus substantially improving the result obtained in [19], which is $$\frac{1}{3}n + 1$$ letters. Additionally, modifying our idea we obtain a family of automata over a three letter alphabet and a subexponential length of the shortest carefully synchronizing words and, as a corollary of that construction, a series of binary automata with a subexponential length of word reducing set of states to a particular subset.