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.

中文翻译：

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同步几乎群自动机

在本文中，我们解决了在几乎群自动机的关键设置中同步随机自动机的问题。群自动机是一种自动机，其中所有字母都充当状态集的排列，并且它们不同步（除非它们具有一个状态）。在几乎群自动机中，其中一个字母充当[公式：见文本]状态的排列，其他字母充当排列。我们证明这个微小的变化足以使自动机以高概率变为同步。更准确地说，我们确定强连通几乎群自动机不同步的概率是 [Formula: see text]，对于​​ [Formula: see text]-字母表。我们还提出了一个有效的算法，它决定一个强连接几乎群自动机正在同步。对于一个自然的计算模型，