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 Theta(3(n/3)) of the shortest carefully synchronizing words, but using 2/9n + 2 letters, thus substantially improving the result obtained in [19], which is 1/3n + 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.