SublinearTime Language Recognition and Decision by OneDimensional Cellular Automata
Abstract
After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (onedimensional) cellular automata: sublineartime 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 sublineartime 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.05828
 Bibcode:
 2019arXiv190905828M
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory
 EPrint:
 16 pages, 2 figures, to appear at DLT 2020