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An “almost dual” to Gottschalk’s Conjecture

Abstract : We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. We then show that on sofic groups, where it is known that injective cellular automata are surjective, post-surjectivity implies pre-injectivity. As no non-sofic groups are currently known, we conjecture that this implication always holds. This mirrors Gottschalk’s conjecture that every injective cellular automaton is surjective.
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Silvio Capobianco, Jarkko Kari, Siamak Taati. An “almost dual” to Gottschalk’s Conjecture. 22th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA), Jun 2016, Zurich, Switzerland. pp.77-89, ⟨10.1007/978-3-319-39300-1_7⟩. ⟨hal-01435035⟩

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