Pre-Expansivity in Cellular Automata

TL;DR

This paper introduces pre-expansivity in cellular automata, exploring its properties, existence, and limitations across different dimensions and types, revealing new distinctions from classical expansivity.

Contribution

It defines pre-expansivity, demonstrates the existence of reversible positively pre-expansive CA in one dimension, and shows non-existence in certain higher-dimensional linear CA.

Findings

Existence of one-dimensional positively pre-expansive CA that are not expansive

Reversible positively pre-expansive CA can exist in 1D

No bi-dimensional linear CA over Abelian groups can be pre-expansive

Abstract

We introduce the property of pre-expansivity for cellular automata (CA): it is the property of being expansive on asymptotic pairs of configurations (i.e. configurations that differ in only finitely many positions). Pre-expansivity therefore lies between expansivity and pre-injectivity, two important notions of CA theory. We show that there exist one-dimensional positively pre-expansive CAs which are not positively expansive and they can be chosen reversible (while positive expansivity is impossible for reversible CAs). We show however that no bi-dimensional CA which is linear over an Abelian group can be pre-expansive. We also consider the finer notion of k-expansivity (expansivity over pairs of configurations with exactly k differences) and show examples of linear CA in dimension 2 and on the free group that are k-expansive depending on the value of k, whereas no (positively) expansive CA exists in this setting.