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Completely Distinguishable Automata and the Set of Synchronizing Words

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Developments in Language Theory (DLT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13911))

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Abstract

An n-state automaton is synchronizing if it can be reset, or synchronized, into a definite state. The set of input words that synchronize the automaton is acceptable by an automaton of size \(2^n - n\). Shortest paths in such an accepting automaton correspond to shortest synchronizing words. Here, we introduce completely distinguishable automata, a subclass of the synchronizing automata. Being completely distinguishable is a necessary condition for a minimal automaton for the set of synchronizing words to have size \(2^n - n\). In fact, as we show, it has size \(2^n - n\) if and only if the automaton is completely distinguishable and has a completely reachable subautomaton that only missed at most one state. We give different characterizations of completely distinguishable automata. Then we relate these notions to graph-theoretical constructions and investigate the subclass of automata with simple idempotents (SI-automata). We show that for these automata the properties of synchronizability, complete distinguishability and complete reachability and the minimal automaton for the set of synchronizing word having \(2^n - n\) states are equivalent when the transformation monoid contains a transitive permutation group. A related result from the literature about SI-automata is wrong, we discuss and correct that mistake here. Lastly, using the results on SI-automata, we show that deciding complete reachability, complete distinguishability and whether the minimal automaton for the set of synchronizing words has \(2^n - n\) states are all \(\textsf{NL}\)-hard problems.

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Acknowledgement

I am thankful to an anonymous referee of a previous version for providing one example from Remark 1 and for having some good suggestions leading to Theorem 2.2 (Item  6). I also thank the referees of the present version for careful reading and identifying some typos. Furthermore, I thank Marek Szykuła & Adam Zyzik for contacting me and telling me about their \(\textsf{PSPACE}\)-completeness result.

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  1. Fachbereich 4 - Abteilung Informatikwissenschaften, Universität Trier, Trier, Germany

    Stefan Hoffmann

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  1. Stefan Hoffmann
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Correspondence to Stefan Hoffmann .

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  1. Umeå University, Umeå, Sweden

    Frank Drewes

  2. Ural Federal University, Ekaterinburg, Russia

    Mikhail Volkov

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Hoffmann, S. (2023). Completely Distinguishable Automata and the Set of Synchronizing Words. In: Drewes, F., Volkov, M. (eds) Developments in Language Theory. DLT 2023. Lecture Notes in Computer Science, vol 13911. Springer, Cham. https://doi.org/10.1007/978-3-031-33264-7_11

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