An “almost dual” to Gottschalk’s Conjecture

  • Silvio CapobiancoEmail author
  • Jarkko Kari
  • Siamak Taati
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9664)


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.


Cellular automata Reversibility Sofic groups 


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Authors and Affiliations

  1. 1.Institute of Cybernetics at Tallinn University of TechnologyTallinnEstonia
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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