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Dynamics of the Independence Number and Automata Synchronization

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Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

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Abstract

We study the lengths of synchronizing words produced by the classical greedy compression algorithm. We associate a sequence of graphs with every synchronizing automaton and rely on evolution of the independence number to bound the lengths of produced words. By leveraging graph theoretical results we show that in many cases automata with good extension properties have good compression properties as well. More precisely, we show that the compression algorithm will produce a synchronizing word of length \(\mathcal {O}(n^2 \log (n))\) on cyclic, regular and strongly-transitive automata with n states, which is not far from the best possible bound of \((n-1)^2\). Furthermore, we provide a relatively simple proof that every n-state automaton has a synchronizing word of length at most \(\frac{n^3}{4} + \mathcal {O}(n^2)\).

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Notes

  1. 1.

    We can even put \(\lceil \tfrac{n-1}{k-1}\rceil \min \{\mathrm {rt}(\mathcal {A}), {n\atopwithdelims ()2}\}\) for any fixed positive integer k.

  2. 2.

    The definitions are provided in Sect. 4.

  3. 3.

    The classical definition also involves initial and final states. Since they are irrelevant for us, we will omit them. Such a model is also often referred to as semiautomaton.

  4. 4.

    Note that this sequence is not assigned to any synchronizing n-state DFA.

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Acknowledgments

We thank Mykhaylo Tyomkyn for fruitful discussions and the reviewers for valuable comments. D. Průša was supported by the Czech Science Foundation under grant number 16-05872S. R. Jungers is a FNRS Research Associate. The work was supported by the French Community of Belgium, the Walloon Region and the Innoviris Foundation. V. Gusev is supported by the Russian Foundation for Basic Research, grant no. 16-01-00795, the Russian Ministry of Education and Science, project no. 1.3253.2017, and the Competitiveness Enhancement Program of Ural Federal University.

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Authors and Affiliations

  1. Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Vladimir V. Gusev & Raphaël M. Jungers

  2. Ural Federal University, Ekaterinburg, Russia

    Vladimir V. Gusev

  3. Czech Technical University, Prague, Czech Republic

    Daniel Průša

Authors
  1. Vladimir V. Gusev
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  2. Raphaël M. Jungers
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  3. Daniel Průša
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Correspondence to Vladimir V. Gusev .

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  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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Gusev, V.V., Jungers, R.M., Průša, D. (2018). Dynamics of the Independence Number and Automata Synchronization. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_31

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  • DOI: https://doi.org/10.1007/978-3-319-98654-8_31

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