Kolmogorov Complexity and the Garden of Eden Theorem

TL;DR

This paper links the Garden of Eden theorem for cellular automata over amenable groups to the preservation of asymptotic Kolmogorov complexity, providing a new characterization of automata with this property.

Contribution

It introduces a novel equivalence between Garden of Eden configurations and Kolmogorov complexity preservation under cellular automata.

Findings

Non-existence of Garden of Eden configurations is equivalent to complexity preservation.

Provides a new characterization of cellular automata that preserve asymptotic Kolmogorov complexity.

Extends the classical Garden of Eden theorem with complexity-theoretic insights.

Abstract

Suppose $\tau$ is a cellular automaton over an amenable group and a finite alphabet. Celebrated Garden of Eden theorem states, that pre-injectivity of $\tau$ is equivalent to non-existence of Garden of Eden configuration. In this paper we will prove, that imposing some mild restrictions, we could add another equivalent assertion: non-existence of Garden of Eden configuration is equivalent to preservation of asymptotic Kolmogorov complexity under the action of cellular automaton. It yields a characterisation of the cellular automata, which preserve the asymptotic Kolmogorov complexity.