Synchronizing Random Almost-Group Automata

TL;DR

This paper proves that almost-group automata, where one letter acts as a permutation on all but one state, are highly likely to be synchronizing, contrasting with group automata which are generally not.

Contribution

It demonstrates that a small modification in automata structure ensures high probability of synchronization in almost-group automata.

Findings

Probability of non-synchronization decreases rapidly with automaton size

Almost-group automata are synchronizing with high probability

Small structural change significantly impacts automaton synchronization

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 $\frac{2^{k-1}-1}{n^{2(k-1)}}(1+o(1))$, for a $k$-letter alphabet.