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On the Interplay Between Babai and Černý’s Conjectures

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Developments in Language Theory (DLT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10396))

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Abstract

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upper-bounded by \(2n^2-6n+5\) and can attain the value \(\tfrac{n(n-1)}{2}\). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter \(\tfrac{n^2}{4} + o(n^2)\).

V. Gusev and M.V. Volkov were supported by RFBR grant no. 16-01-00795, Russian Ministry of Education and Science project no. 1.3253.2017, and the Competitiveness Enhancement Program of Ural Federal University. Balázs Gerencsér was supported by PD grant no. 121107, National Research, Development and Innovation Office of Hungary. This work was supported by the French Community of Belgium and by the IAP network DYSCO. Raphaël Jungers is a Fulbright Fellow and a FNRS Research Associate.

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Notes

  1. 1.

    As initial and final states play no role in our considerations, we omit them.

  2. 2.

    During the preparation of this paper we discovered that the same question was also posed in [34, Conjecture 3], though its connection with Babai’s problem was not registered there.

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Authors and Affiliations

  1. ICTEAM Institute, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    François Gonze, Vladimir V. Gusev & Raphaël M. Jungers

  2. Ural Federal University, Ekaterinburg, Russia

    Vladimir V. Gusev & Mikhail V. Volkov

  3. Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Balázs Gerencsér

Authors
  1. François Gonze
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  2. Vladimir V. Gusev
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  3. Balázs Gerencsér
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  4. Raphaël M. Jungers
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  5. Mikhail V. Volkov
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Correspondence to François Gonze .

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  1. Université de Liège, Liège, Belgium

    Émilie Charlier

  2. Université de Liège, Liège, Belgium

    Julien Leroy

  3. Université de Liège, Liège, Belgium

    Michel Rigo

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Gonze, F., Gusev, V.V., Gerencsér, B., Jungers, R.M., Volkov, M.V. (2017). On the Interplay Between Babai and Černý’s Conjectures. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_13

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  • DOI: https://doi.org/10.1007/978-3-319-62809-7_13

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