On reversible asynchronous non-uniform cellular automata

TL;DR

This paper explores the properties of asynchronous non-uniform cellular automata, establishing key equivalences and invertibility conditions, and highlighting differences from traditional cellular automata.

Contribution

It introduces the concepts of stable injectivity, stable reversibility, and stable post-surjectivity for ANUCA, and proves their equivalence and invertibility in various classes.

Findings

Reversibility, stable reversibility, and stable injectivity are equivalent for ANUCA.

Several classes of injective and stably injective ANUCA are invertible.

Counter-examples demonstrate differences between cellular automata and ANUCA.

Abstract

We study the class of asynchronous non-uniform cellular automata (ANUCA) over an arbitrary group universe with multiple local transition rules. We introduce the notion of stable injectivity, stable reversibility, stable post-surjectivity and investigate several dynamical properties of such automata. In particular, we establish the equivalence between reversibility, stable reversibility, and stable injectivity for ANUCA. We also prove the invertibility of several classes of injective and stably injective ANUCA. Counter-examples are given to highlight the differences between cellular automata and ANUCA.