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Register a new userLower Bounds for Synchronizing Word Lengths in Partial Automata
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Michiel de Bondt
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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 this bound was only given for $n leq 5$, and with bounds on the number of symbols for $n leq 12$. Here we give the full analysis for $n leq 7$, without bounds on the number of symbols. For PFAs (partial automata) on $leq 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n geq 4$. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on $leq 10$ states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of $n$. For $n=6,7,8,9$, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than $(n-1)^2$, although still quadratic. Based on string rewriting, for arbitrary size we construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation. Both PFAs are transitive. Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.
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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 $mathcal{C}_n$ have switch count linear in $n$, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one reaching $frac{2}{3} n^2 + O(n)$ as switch count. We investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all $3leq nleq 9$, a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every $n$, no automata are known with higher synchronization length than $mathcal{C}_n$, which has only two symbols. It is not clear if this pattern extends to larger $n$. For $ngeq 12$, our best construction only has two symbols.
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 result of [1-3] for the emptiness problem of weakly synchronizing language is incorrect.
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 forms an upper bound for this class and we experimentally verify it for small automata by an exhaustive computation.
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 role of the alphabet size. For each possible alphabet size, we count DFAs on $n le 6$ states which synchronize in $(n-1)^2 - e$ steps, for all $e < 2lceil n/2 rceil$. Furthermore, we give constructions of automata with any number of states, and $3$, $4$, or $5$ symbols, which synchronize slowly, namely in $n^2 - 3n + O(1)$ steps. In addition, our results prove v{C}ernys conjecture for $n le 6$. Our computation has led to $27$ DFAs on $3$, $4$, $5$ or $6$ states, which synchronize in $(n-1)^2$ steps, but do not belong to v{C}ernys sequence. Of these $27$ DFAs, $19$ are new, and the remaining $8$ which were already known are exactly the emph{minimal} ones: they will not synchronize any more after removing a symbol. So the $19$ new DFAs are extensions of automata which were already known, including the v{C}erny automaton on $3$ states. But for $n > 3$, we prove that the v{C}erny automaton on $n$ states does not admit non-trivial extensions with the same smallest synchronizing word length $(n-1)^2$.
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.
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