Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata
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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 cellular automata 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 respective to the parallel computation classes SC and (uniform) AC. As an addendum, we introduce and investigate the concept of a strong ACA (SACA) as the decider counterpart of a (weak) ACA, show the class of languages decidable in constant time by SACAs equals the locally testable languages, and determine Ω(√(n)) as the (tight) time complexity threshold for SACAs up to which no advantage respective to constant time is possible.
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