We study the generic limit sets of one-dimensional cellular automata, which intuitively capture their asymptotic dynamics while discarding transient phenomena. As our main results, we characterize the automata whose generic limit set is a singleton, and show that this class is $\Sigma^0_2$-complete. We also prove that given a CA whose generic limit set is guaranteed to be a singleton, the sole configuration it contains cannot be algorithmically determined.

中文翻译：

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一般幂零元胞自动机

我们研究了一维元胞自动机的通用极限集，它直观地捕捉了它们的渐近动力学，同时丢弃了瞬态现象。作为我们的主要结果，我们描述了通用限制集是单例的自动机，并表明这个类是 $\Sigma^0_2$-complete。我们还证明，给定一个 CA，其通用限制集保证是单例，它包含的唯一配置无法通过算法确定。