Reversibility, balance and expansivity of non-uniform cellular automata

TL;DR

This paper investigates properties like reversibility, balance, and expansivity in 1D non-uniform cellular automata, revealing conditions under which these properties hold and providing examples illustrating their complex behaviors.

Contribution

It establishes new conditions linking rule recurrence to reversibility and balance in NUCA, and explores the dynamics of expansive and equicontinuous NUCA.

Findings

Bijective NUCA with recurrent rule distribution are reversible.

Surjective or bijective NUCA with recurrent rules are balanced.

Expansive NUCA are sensitive.

Abstract

Non-uniform cellular automata (NUCA) are an extension of cellular automata (CA), which transform cells according to multiple different local rules. A NUCA is defined by a configuration of local rules called a local rule distribution. We examine what properties of uniform CA can be recovered by restricting the rule distribution to be (uniformly) recurrent, focusing on only 1D NUCA. We show that a bijective NUCA with a uniformly recurrent rule distribution is reversible. We also show that if a NUCA is surjective and has a recurrent rule distribution, or if it is bijective, then it is balanced. We present an example of a NUCA which has a non-empty and non-residual set of equicontinuity points, and one which is not sensitive but has no equicontinuity points. Finally, we show that (positively) expansive NUCA are sensitive.