Complexities of some problems related to synchronizing, nonsynchronizing and monotonic automata

In this study, we first introduce several problems related to finding reset words for deterministic finite automata, and present motivations for these problems for practical applications in areas such as robotics and bio-engineering. We then analyse computational complexities of these problems. Second, we consider monotonic and partially specified automata. Monotonicity is known to be a feature simplyfing the synchronizability problems. On the other hand for partially specified automata, synchronizability problems are known to be harder than the completely specified automata. We investigate the complexity of some synchronizability problems for automata that are both monotonic and partially specified. We show that checking the existence, computing one, and computing a shortest reset word for a monotonic partially specified automaton is NP-hard. We also show that finding a reset word that synchronizes $\mathcal{K}$ number of states (or maximum number of states) of a given monotonic non-synchronizable automaton to a given set of states is NP-hard.