Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

Augusto Modanese

doi:10.1007/978-3-030-48516-0_19

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arXiv · 1909.05828

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

Augusto Modanese

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Abstract

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.

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Augusto Modanese. 2019-09-12. Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata. https://doi.org/10.1007/978-3-030-48516-0_19

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