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We consider the problem {sc Max Sync Set} of finding a maximum synchronizing set of states in a given automaton. We show that the decision version of this problem is PSPACE-complete and investigate the approximability of {sc Max Sync Set} for binary and weakly acyclic automata (an automaton is called weakly acyclic if it contains no cycles other than self-loops). We prove that, assuming $P e NP$, for any $varepsilon > 0$, the {sc Max Sync Set} problem cannot be approximated in polynomial time within a factor of $O(n^{1 - varepsilon})$ for weakly acyclic $n$-state automata with alphabet of linear size, within a factor of $O(n^{frac{1}{2} - varepsilon})$ for binary $n$-state automata, and within a factor of $O(n^{frac{1}{3} - varepsilon})$ for binary weakly acyclic $n$-state automata. Finally, we prove that for unary automata the problem becomes solvable in polynomial time.
Instead of looking at the lengths of synchronizing words as in v{C}ernys conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the v{C}erny automata $ma
We present an infinite series of $n$-state Eulerian automata whose reset words have length at least $(n^2-3)/2$. This improves the current lower bound on the length of shortest reset words in Eulerian automata. We conjecture that $(n^2-3)/2$ also for
In [1], we introduced the weakly synchronizing languages for probabilistic automata. In this report, we show that the emptiness problem of weakly synchronizing languages for probabilistic automata is undecidable. This implies that the decidability re
It was conjectured by v{C}erny in 1964 that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. In this paper, we investigate the ro
It was conjectured by v{C}erny in 1964, that a synchronizing DFA on $n$ states always has a synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching