We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one- and two-sided error versions of the model (in the same spirit as the classes $\mathsf{RP}$ and $\mathsf{BPP}$) and establish a separation between the classes of languages they can recognize all the way up to $o(\sqrt{n})$ time. As a consequence, we have a $\Omega(\sqrt{n})$ lower bound for derandomizing constant-time two-sided error PACAs (using deterministic ACAs). We also prove that derandomization of $T(n)$-time PACAs (to polynomial-time deterministic cellular automata) for various regimes of $T(n) = \omega(\log n)$ implies non-trivial derandomization results for the class $\mathsf{RP}$ (e.g., $\mathsf{P} = \mathsf{RP}$). The main contribution is an almost full characterization of the constant-time PACA classes: For one-sided error, the class equals that of the deterministic model; that is, constant-time one-sided error PACAs can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we identify a natural class we call the linearly testable languages ($\mathsf{LLT}$) and prove that the languages decidable by constant-time two-sided error PACAs are "sandwiched" in-between the closure of $\mathsf{LLT}$ under union and intersection and the class of locally threshold testable languages ($\mathsf{LTT}$).

翻译：我们提出并研究了一种亚线性时间一维概率元胞自动机模型。具体而言，我们改进了ACA模型（一种当且仅当所有元胞同时接受时才接受输入的元胞自动机），使得每个元胞的状态更新不仅依赖于其邻域内观察到的状态，还依赖于自身抛掷的无偏硬币结果。由此产生的模型称为概率ACA（PACA）。我们考虑了该模型的单侧误差和双侧误差版本（与复杂度类$\mathsf{RP}$和$\mathsf{BPP}$的对应概念一致），并建立了它们在$o(\sqrt{n})$时间范围内可识别语言类别之间的分离关系。由此产生的结论是，对于使用确定性ACA的常数时间双侧误差PACA去随机化，存在$\Omega(\sqrt{n})$的下界。我们还证明，对于$T(n) = \omega(\log n)$的不同规模，$T(n)$时间PACA（到多项式时间确定性元胞自动机）的去随机化蕴含了复杂度类$\mathsf{RP}$的非平凡去随机化结果（例如$\mathsf{P} = \mathsf{RP}$）。主要贡献在于对常数时间PACA类别的近乎完全刻画：对于单侧误差，该类等同于确定性模型类别，即常数时间单侧误差PACA可以在时间复杂度上仅以常数乘法开销实现完全去随机化。至于双侧误差，我们识别出一个称为线性可测试语言（$\mathsf{LLT}$）的自然类别，并证明可由常数时间双侧误差PACA判定的语言“夹在”$\mathsf{LLT}$关于并集和交集闭包与局部阈值可测试语言（$\mathsf{LTT}$）类别之间。