A converse to Moore's theorem on cellular automata

TL;DR

This paper establishes a new characterization of group amenability through cellular automata, proving that amenability is equivalent to the existence of gardens of Eden in automata with mutually erasable patterns.

Contribution

It proves the first part of a conjecture linking group amenability to cellular automata properties, specifically the existence of gardens of Eden.

Findings

Group G is amenable if and only if all cellular automata with mutually erasable patterns on G have gardens of Eden.

Confirmed a conjecture relating amenability to cellular automata behavior.

Provides a new criterion for amenability based on cellular automata properties.

Abstract

We prove a converse to Moore's ``Garden-of-Eden'' theorem: a group G is amenable if and only if all cellular automata living on G that admit mutually erasable patterns also admit gardens of Eden. It had already been conjectured in that amenability could be characterized by cellular automata. We prove the first part of that conjecture.