Advertisement

An “almost dual” to Gottschalk’s Conjecture

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

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 

References

  1. 1.
    Bartholdi, L.: Gardens of Eden and amenability on cellular automata. J. Eur. Math. Soc. 12(1), 241–248 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Capobianco, S.: On the induction operation for shift subspaces and cellular automata as presentations of dynamical systems. Inform. Comput. 207(11), 1169–1180 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Capobianco, S., Guillon, P., Kari, J.: Surjective cellular automata far from the Garden of Eden. Disc. Math. Theor. Comput. Sci. 15(3), 41–60 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Capobianco, S., Kari, J., Taati, S.: Post-surjectivity and balancedness of cellular automata over groups. In: Kari, J., Törmä, I., Szabados, M. (eds) 21st International Workshop on Cellular Automata and Discrete Complex Systems: Exploratory Papers of AUTOMATA 2015, Turku Centre for Computer Science, TUCS Lecture Notes, vol. 24, pp. 31–38 (2015)Google Scholar
  5. 5.
    Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups. Springer Monographs in Mathematics. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ceccherini-Silberstein, T., Machì, A., Scarabotti, F.: Amenable groups and cellular automata. Ann. Inst. Fourier 49(2), 673–685 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiorenzi, F.: Cellular automata and strongly irreducible shifts of finite type. Theor. Comput. Sci. 299, 477–493 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gottschalk, W.H.: Some general dynamical notions. In: Gottschalk, W. (ed.) Recent Advances in Topological Dynamics. Lecture Notes in Mathematics, vol. 318. Springer, Heidelberg (1973)CrossRefGoogle Scholar
  9. 9.
    Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. European Math. Soc. 1, 109–197 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kari, J.: Theory of cellular automata: a survey. Theor. Comp. Sci. 334, 3–33 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kari, J., Taati, S.: Statistical mechanics of surjective cellular automata. J. Stat. Phys. 160(5), 1198–1243 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  13. 13.
    Weiss, B.: Sofic groups and dynamical systems. Sankhyā: Indian. J Stat. 62A(3), 350–359 (2000)zbMATHGoogle Scholar

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

Personalised recommendations