Kolmogorov Complexity and the Garden of Eden Theorem

The Garden of Eden theorem: old and new

We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.

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

Constrained Eden

Let CA be a one-dimensional cellular automaton. A computational complexity problem, the Constrained Eden problem, denoted C-EDEN(CA), is introduced. C-EDEN(CA) is a finitary variant of the Garden of Eden problem. Even for certain elementary cellular automata, C-EDEN(CA) is NP-complete, providing the first examples of NP-complete problems associated with cellular automata over a twoelement alpha...