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

TL;DR

This paper explores sublinear-time computation in one-dimensional cellular automata, establishing hierarchy results, analyzing their relation to regular languages, and introducing a new decider model with specific complexity thresholds.

Contribution

It proves a time hierarchy for sublinear ACA classes, analyzes their intersection with regular languages, and introduces the decider ACA model with its complexity properties.

Findings

Established a time hierarchy theorem for sublinear ACA classes.

Analyzed the intersection of ACA classes with regular languages.

Introduced and characterized the decider ACA (DACA) model and its complexity thresholds.

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.