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.
Capobianco, S., Kari, J., & Taati, S. (2016). An “almost dual” to Gottschalk’s conjecture. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9664, pp. 77–89). Springer Verlag. https://doi.org/10.1007/978-3-319-39300-1_7