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arXiv · 1704.04047

On the interplay between Babai and Cerny's conjectures

François Gonze·Vladimir Gusev·Balázs Gerencsér·Raphaël M. Jungers·Mikhail V. Volkov
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Abstract

Motivated by the Babai conjecture and the Cerny 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 $\tfrac{n^2}{4} + o(n^2)$.

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François Gonze, Vladimir Gusev, Balázs Gerencsér, Raphaël M. Jungers, Mikhail V. Volkov. 2017-04-13. On the interplay between Babai and Cerny's conjectures. https://arxiv.org/abs/1704.04047

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