The Garden of Eden theorem: old and new

Kolmogorov Complexity and the Garden of Eden Theorem

Suppose τ is a cellular automaton over an amenable group and a finite alphabet. Celebrated Garden of Eden theorem states, that pre-injectivity of τ 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 preservatio...

Surjective cellular automata far from the Garden of Eden

One of the first and most famous results of cellular automata theory, Moore’s Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in d...

Some Extensions of the Moore-Myhill Garden of Eden Theorem

Cellular Automata in Non-Euclidean Spaces

Classical results on the surjectivity and injectivity of parallel maps are shown to be extendible to the cases with non-Euclidean cell spaces of particular types. Also shown are obstructions to extendibility, which may shed light on the nature of classical results such as the Garden-of-Eden theorem. Key–Words: Cellular automata, Non Euclidean cell spaces, the Garden-of-Eden theorem

Amenability of groups is characterized by Myhill's Theorem

We prove a converse to Myhill’s “Garden-of-Eden” theorem and obtain in this manner a characterization of amenability in terms of cellular automata: A group G is amenable if and only if every cellular automaton with carrier G that has gardens of Eden also has mutually erasable patterns. This answers a question by Schupp, and solves a conjecture by CeccheriniSilberstein, Mach̀ı and Scarabotti.