Kolmogorov Complexity and the Garden of Eden Theorem

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.