On the Interplay between Aerný and Babai's Conjectures

Abstract

Motivated by the Babai conjecture and the Aerný 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 threshold is upper-bounded by 2n2 - 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 n2 4 + o(n2). © 2019 World Scientific Publishing Company.