Synchronizing Random Almost-Group Automata

Abstract

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on n − 1 states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability.More precisely, we establish that the probability that a strongly connected almost-group automaton is not synchronizing is 2 k−1 −1 n 2(k−1) (1 + o(1)), for a k-letter alphabet.