Synchronizing Almost-Group Automata

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 [Formula: see text] 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 [Formula: see text], for a [Formula: see text]-letter alphabet. We also present an efficient algorithm that decides whether a strongly-connected almost-group automaton is synchronizing. For a natural model of computation, we establish a [Formula: see text] worst-case lower bound for this problem ([Formula: see text] for the average case), which is almost matched by our algorithm.

Synchronizing random automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.

RANDOMIZED CELLULAR AUTOMATA

We define and study a few properties of a class of random automata networks. While regular finite one-dimensional cellular automata are defined on periodic lattices, these automata networks, called randomized cellular automata, are defined on random directed graphs with constant out-degrees and evolve according to cellular automaton rules. For some families of rules, a few typical a priori unexpected results are presented.