Some pointwise and decidable properties of non-uniform cellular automata

TL;DR

This paper investigates properties of non-uniform cellular automata, establishing equivalences between pointwise and global properties, and proving decidability results for various classes of NUCA, including linear and perturbed automata.

Contribution

It introduces new equivalences between pointwise and global properties of NUCA and proves decidability for these properties in linear and perturbed cases.

Findings

Pointwise nilpotency, periodicity, and eventual periodicity are equivalent to their global counterparts.

Linear NUCA satisfying polynomial equations are eventually periodic.

Decidability results are established for NUCA with finite memory and local perturbations.

Abstract

For non-uniform cellular automata (NUCA) with finite memory over an arbitrary universe with multiple local transition rules, we show that pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity properties are respectively equivalent to nilpotency, periodicity, and eventual periodicity. Moreover, we prove that every linear NUCA which satisfies pointwise a polynomial equation (which may depend on the configuration) must be an eventually periodic linear NUCA. Generalizing results for higher dimensional group and linear CA, we also establish the decidability results of the above dynamical properties as well as the injectivity for arbitrary NUCA with finite memory which are local perturbations of higher dimensional linear and group CA. Some generalizations to the case of sparse global perturbations of higher dimensional linear and group CA are also obtained.