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

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 [Formula: see text] and (uniform) [Formula: see text]. 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 [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata

“…From the perspective of cellular automata theory, our work furthers knowledge in sublineartime cellular automata models, a topic seemingly neglected by the CA community at large (as discussed in, e.g., [19]). Although this is certainly not the first result in which complexitytheoretical results for cellular automata and their variants have consequences for classical models (see, e.g., [15,24] for results in this sense), to the best of our knowledge said results address only necessary conditions for separating classical complexity classes.…”

“…These limitations hold not only for SCAs but also for standard CAs. Nevertheless, SCAs are more powerful than other CA models capable of sublinear-time computation such as ACAs [11,19], which are CAs with their acceptance behavior such that the CA accepts if and only if all cells simultaneously accept. This is because SCAs can efficiently aggregate results computed in parallel (by combining them using some efficiently computable function); in ACAs any such form of aggregation is fairly limited as the underlying cell structure is static.…”

“…The concept of block words seems to arise naturally in the context of sublinear-time (both shrinking and standard) cellular automata [11,19]. The syntax of block words is very efficiently verifiable (more precisely, in time linear in the block length) by a CA (without need of shrinking).…”

“…Coupling block words with cellular automata yields a computational paradigm that appears to be substantially diverse from linear-and real-time cellular automata computation (see [19] for examples). Often we shall describe operations on a block (rather than on a cell) level and, by making use of block numbering, two blocks with distinct numbers may operate differently even though their contents are the same; this would be impossible at a cell level due to the locality of CA rules.…”

“…For example, An SCA can verify its input is a valid block word in O(b(n)) time, where with "verify" we mean every block checks locally that its structure and contents are consistent (with Definition 8). This can be realized using standard CA techniques and, in particular, does not require shrinking; see, for instance, [11,19] for detailed constructions. Recall that in Definition 5 we did not require an SCA S to explicitly reject inputs not in L(S); hence, the time complexity of S on an input w counts only towards the complexity of S if w ∈ L(S), that is, the time complexity of S on any w ∈ L(S) is ignored.…”