A Probabilistic Cellular Automata Rule Forming Domino Patterns

  • Rolf HoffmannEmail author
  • Dominique Désérable
  • Franciszek Seredyński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11657)


The objective in this study is to form a domino pattern by Cellular Automata (CA). In a previous work such patterns were formed by CA agents, which were trained with high effort by the aid of Genetic Algorithm. Now two probabilistic CA rules are designed in a methodical way that can perform this task very reliably even for rectangular fields. The first rule evolves stable sub–optimal pattern. The second rule maximizes the overlap between dominoes thereby maximizing the number of dominoes.


Pattern formation Probabilistic cellular automata Matching templates Asynchronous updating Parallel Substitution Algorithm 


  1. 1.
    Chopard, B., Droz, M.: Cellular Automata Modeling of Physical Systems. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  2. 2.
    Deutsch, A., Dormann, S.: Cellular Automaton Modeling of Biological Pattern Formation. Birkäuser (2005)Google Scholar
  3. 3.
    Désérable, D., Dupont, P., Hellou, M., Kamali-Bernard, S.: Cellular automata in complex matter. Complex Syst. 20(1), 67–91 (2011)MathSciNetGoogle Scholar
  4. 4.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55(3), 601–644 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Nagpal, R.: Programmable pattern-formation and scale-independence. In: Minai, A.A., Bar-Yam, Y. (eds.) Unifying Themes in Complex Sytems IV, pp. 275–282. Springer, Heidelberg (2008). Scholar
  6. 6.
    Yamins, D., Nagpal, R.: Automated Global-to-Local programming in 1-D spatial multi-agent systems. In: Proceedings 7th International Conference on AAMAS, pp. 615–622 (2008)Google Scholar
  7. 7.
    Tomassini, M., Venzi, M.: Evolution of asynchronous cellular automata for the density task. In: Guervós, J.J.M., Adamidis, P., Beyer, H.-G., Schwefel, H.-P., Fernández-Villacañas, J.-L., (eds.): Parallel Problem Solving from Nature – PPSN VIIPPSN 2002. LNCS, vol. 2439, pp. 934–943. Springer, Heidelberg (2002). Scholar
  8. 8.
    Birgin, E.G., Lobato, R.D., Morabito, R.: An effective recursive partitioning approach for the packing of identical rectangles in a rectangle. J. Oper. Research Soc. 61, 303–320 (2010)CrossRefGoogle Scholar
  9. 9.
    Hoffmann, R.: How agents can form a specific pattern. In: Wąs, J., Sirakoulis, G.C., Bandini, S. (eds.) ACRI 2014. LNCS, vol. 8751, pp. 660–669. Springer, Cham (2014). Scholar
  10. 10.
    Hoffmann, R.: Cellular automata agents form path patterns effectively. Acta Phys. Pol. B Proc. Suppl. 9(1), 63–75 (2016)CrossRefGoogle Scholar
  11. 11.
    Hoffmann, R., Désérable, D.: Line patterns formed by cellular automata agents. In: El Yacoubi, S., Wąs, J., Bandini, S. (eds.) ACRI 2016. LNCS, vol. 9863, pp. 424–434. Springer, Cham (2016). Scholar
  12. 12.
    Hoffmann, R., Désérable, D.: Generating maximal domino patterns by cellular automata agents. In: Malyshkin, V. (ed.) PaCT 2017. LNCS, vol. 10421, pp. 18–31. Springer, Cham (2017). Scholar
  13. 13.
    Achasova, S., Bandman, O., Markova, V., Piskunov, S.: Parallel Substitution Algorithm, Theory and Application. World Scientific, Singapore (1994)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rolf Hoffmann
    • 1
    Email author
  • Dominique Désérable
    • 2
  • Franciszek Seredyński
    • 3
  1. 1.Technische Universität DarmstadtDarmstadtGermany
  2. 2.Institut National des Sciences AppliquéesRennesFrance
  3. 3.Department of Mathematics and Natural SciencesCardinal Stefan Wyszynski UniversityWarsawPoland

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