Sublinear-Time Probabilistic Cellular Automata

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. We also prove that the derandomization of $T(n)$-time PACA (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 is equal to that of the deterministic model; that is, constant-time one-sided error PACA can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we prove that the respective class is "sandwiched" in-between the class of strictly locally testable languages ($\mathsf{SLT}$) and that of locally threshold testable languages ($\mathsf{LTT}$).