A goal of this paper is to introduce the new construction of an automaton with shortest synchronizing word of length $O(d^{ \frac{ n}{ d}})$, where $d \in \mathbb{ N}$ and $n$ is the number of states for that automaton. Additionally we introduce new transformation from any synchronizable DFA or carefully synchronizable PFA of $n$ states to carefully synchronizable PFA of $d \cdot n$ states with shortest synchronizing word of length $\Omega(d^{ \frac{ n}{ d}})$.
Categories: cs.FL cs.DM