On the surjunctivity and the Garden of Eden theorem for non-uniform cellular automata

TL;DR

This paper investigates conditions under which the Garden of Eden theorem applies to non-uniform cellular automata, focusing on rule distributions that are recurrent or asymptotic to recurrent distributions, and explores surjunctivity in this context.

Contribution

It establishes the validity of the Garden of Eden theorem for NUCA with recurrent rule distributions and introduces conditions for surjunctivity based on asymptotic rule distributions.

Findings

Garden of Eden theorem holds for NUCA with recurrent rule distributions.

Non-recurrent rule distributions can produce NUCA violating the theorem.

Asymptotic to recurrent distributions imply surjunctive NUCA.

Abstract

Non-uniform cellular automata (NUCA) are an extension of cellular automata with multiple local rules in different cells. We show that if the distribution of local rules is uniformly recurrent, or recurrent in the one-dimensional case, the Garden of Eden theorem holds. We also show that for any non-recurrent distribution, there is a substitution of local rules that defines a NUCA which does not satisfy the Garden of Eden theorem. Finally, we show that a rule distribution asymptotic to recurrent distribution defines a surjunctive NUCA.