After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes $\mathsf{SC}$ and (uniform) $\mathsf{AC}$. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine $\Omega(\sqrt{n})$ as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

中文翻译：

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一维元胞自动机的亚线性时间语言识别与决策

经过大约 30 年的明显中断，我们重新审视了（一维）元胞自动机理论中一个看似被忽视的主题：次线性时间计算。考虑的模型是 ACA 模型，它是语言接受器，其接受条件取决于自动机中所有单元格的状态。我们证明了次线性时间 ACA 类的时间层次定理，分析它们与常规语言的交集，最后在并行计算类 $\mathsf{SC}$ 和（统一）$\mathsf{AC} 中建立严格包含$. 作为附录，我们介绍并研究了决策者 ACA (DACA) 的概念，作为（接受者）ACA 的决策者对应物的候选者。我们证明 DACA 在恒定时间内可判定的语言类等于本地可测试的语言，