An “almost dual” to Gottschalk’s Conjecture

Conference paper

DOI: 10.1007/978-3-319-39300-1_7

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9664)
Cite this paper as:
Capobianco S., Kari J., Taati S. (2016) An “almost dual” to Gottschalk’s Conjecture. In: Cook M., Neary T. (eds) Cellular Automata and Discrete Complex Systems. AUTOMATA 2016. Lecture Notes in Computer Science, vol 9664. Springer, Cham

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.

Keywords

Cellular automata Reversibility Sofic groups 

Copyright information

© IFIP International Federation for Information Processing 2016

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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