On the interplay between Babai and \v{C}ern\'y's conjectures

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 upperbounded by 2n − 6n + 5 and can attain the value n(n−1) 2 . In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter n 2 4 + o(n). 1 Background and Overview Completely reachable automata, i.e., deterministic finite automata in which every non-empty subset of the state set occurs as the image of the whole state set under the action of a suitable input word, appeared in the study of descriptional complexity of formal languages [26] and in relation to the Černý conjecture [13]. In [6] an emphasis has been made on automata in this class with minimal transition monoid size. In the present paper we focus on automata being in a sense the extreme opposites of those studied in [6], namely, on automata of maximal transition monoid size. In other words, we consider automata with full transition monoid, i.e., transition monoid equal to the full monoid of transformations of the state set; clearly, automata with this property are completely reachable. There are several reasons justifying special attention to automata with full transition monoid. First, as observed in [6], the membership problem for this class of automata is decidable in polynomial time (of the size of the input automaton) while ⋆ Vladimir Gusev and Mikhail V. Volkov were supported by RFBR grant no. 16-0100795, 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. A short version of this work has been presented at the conference DLT 2017.