Discrete Mathematics & Theoretical Computer Science, Vol 15, No 3 (2013)

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Surjective cellular automata far from the Garden of Eden

Silvio Capobianco, Pierre Guillon, Jarkko Kari


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 dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.

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