Equicontinuity and Sensitivity of Nondeterministic Cellular Automata

  • Pietro Di LenaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10248)


Nondeterministic Cellular Automata (NCA) are the class of multivalued functions characterized by nondeterministic block maps. We extend the notions of equicontinuity and sensitivity to multivalued functions and investigate the characteristics of equicontinuous, almost equicontinuous and sensitive NCA. The dynamical behavior of nondeterministic CA in these classes is much less constrained than in the deterministic setting. In particular, we show that there are transitive NCA with equicontinuous points and equicontinuous NCA that are not reversible.


Nondeterministic cellular automata Equicontinuity Sensitivity Transitivity 


  1. 1.
    Berge, C.: Topological Spaces. Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. Dover Publications Inc., Mineola (1963)zbMATHGoogle Scholar
  2. 2.
    Burkhead, E., Hawkins, J.M.: Nondeterministic and stochastic cellular automata and virus dynamics. PreprintGoogle Scholar
  3. 3.
    Dennunzio, A., Di Lena, P., Formenti, E., Margara, L.: On the directional dynamics of additive cellular automata. Theoret. Comput. Sci. 410, 4823–4833 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dennunzio, A., Formenti, E., Manzoni, L., Mauri, G.: m-Asynchronous cellular automata: from fairness to quasi-fairness. Natural Comput. 12, 561–572 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dennunzio, A., Formenti, E., Provillard, J.: Local rule distributions, language complexity and non-uniform cellular automata. Theor. Comput. Sci. 504, 38–51 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dennunzio, A., Di Lena, P., Formenti, E., Margara, L.: Periodic orbits and dynamical complexity in cellular automata. Fundam. Inform. 126, 183–199 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dennunzio, A., Formenti, E., Weiss, M.: Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues. Theor. Comput. Sci. 516, 40–59 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dennunzio, A., Formenti, E., Manzoni, L., Mauri, G., Porreca, A.E.: Computational complexity of finite asynchronous cellular automata. Theor. Comput. Sci. 664, 131–143 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Di Lena, P.: Decidable properties for regular cellular automata. In: Fourth IFIP International Conference on Theoretical computer Science, pp. 185–196 (2006)Google Scholar
  10. 10.
    Di Lena, P., Margara, L.: Computational complexity of dynamical systems: the case of cellular automata. Inform. Comput. 206, 1104–1116 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Di Lena, P., Margara, L.: On the undecidability of the limit behavior of Cellular Automata. Theoret. Comput. Sci. 411, 1075–1084 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Di Lena, P., Margara, L.: On the undecidability of attractor properties for cellular automata. Fund. Inform. 115, 78–85 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Di Lena, P., Margara, L.: Nondeterministic cellular automata. Inform. Sci. 287, 13–25 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Furusawa, H., Ishida, T., Kawahara, Y.: Continuous relations and Richardson’s Theorem. In: Kahl, W., Griffin, T.G. (eds.) RAMICS 2012. LNCS, vol. 7560, pp. 310–325. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33314-9_21 CrossRefGoogle Scholar
  15. 15.
    Furusawa, H.: Uniform continuity of relations and nondeterministic cellular automata. Theoret. Comput. Sci. 673, 19–29 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kůrka, P.: Languages, equicontinuity and attractors in cellular automata. Ergod. Theor. Dyn. Syst. 17, 417–433 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kůrka, P.: Topological and Symbolic Dynamics. Cours Spécialisés 11, Société Mathématique de France, Paris (2003)Google Scholar
  19. 19.
    Ozhigov, Y.: Computations on nondeterministic cellular automata. Inform. Comput. 148, 181–201 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Richardson, D.: Tessellations with local transformations. J. Comput. Syst. Sci. 6, 373–388 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yaku, T.: Surjectivity of nondeterministic parallel maps induced by nondeterministic cellular automata. J. Comput. Syst. Sci. 12, 1–5 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

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© IFIP International Federation for Information Processing 2017

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of BolognaBolognaItaly

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