Progresses in the Analysis of Stochastic 2D Cellular Automata: A Study of Asynchronous 2D Minority

  • Damien Regnault
  • Nicolas Schabanel
  • Éric Thierry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)


Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on one-dimensional probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from one- to two-dimensional systems is a long lasting challenging problem.

In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.

Due to space constraint, we refer the reader to [16] for the missing proofs.


Convex Hull Cellular Automaton Minority State Checkerboard Pattern Transition Table 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aracena, J., Lamine, S.B., Mermet, M.-A., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Trans. on Systems, Man, and Cybernetics Part B 33(5), 825–834 (2003)Google Scholar
  2. 2.
    Bersini, H., Detours, V.: Asynchrony induces stability in cellular automata based models. In: Proceedings of Artificial Life IV, pp. 382–387. MIT Press, Cambridge (1994)Google Scholar
  3. 3.
    Bovier, A., Manzo, F.: Metastability in glauber dynamics in the low temperature limit: Beyond exponential asymptotics. J. Statist. Phys. 107, 757–779 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buvel, R.L., Ingerson, T.E.: Structure in asynchronous cellular automata. Physica D 1, 59–68 (1984)MathSciNetGoogle Scholar
  5. 5.
    Fatès, N., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. Complex Systems 16(1), 1–27 (2005)MathSciNetGoogle Scholar
  6. 6.
    Fatès, N., Morvan, M., Schabanel, N., Thierry, É.: Fully asynchronous behaviour of double-quiescent elementary cellular automata. Theoretical Computer Science 362, 1–16 (2006) (An extended abstract was also published in Proc. of MFCS’2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fatès, N., Regnault, D., Schabanel, N., Thierry, É.: Asynchronous behaviour of double-quiescent elementary cellular automata. In: LATIN 2006. LNCS, vol. 3887, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Fukś, H.: Non-deterministic density classification with diffusive probabilistic cellular automata. Phys. Rev. E 66(2) (2002)Google Scholar
  9. 9.
    Fukś, H.: Probabilistic cellular automata with conserved quantities. Nonlinearity 17(1), 159–173 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goles, E., Martinez, S.: Neural and automata networks, dynamical behavior and applications. In: Maths and Applications, vol. 58, Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  11. 11.
  12. 12.
  13. 13.
    Lumer, E.D., Nicolis, G.: Synchronous versus asynchronous dynamics in spatially distributed systems. Physica D 71, 440–452 (1994)zbMATHCrossRefGoogle Scholar
  14. 14.
    Randall, D.: Mixing. In: Proc. of the Symp. on Foundations of Computer Science (FOCS), pp. 4–15 (2003)Google Scholar
  15. 15.
    Regnault, D.: Abrupt behaviour changes in cellular automata under asynchronous dynamics. In: Proceedings of 2nd European Conference on Complex Systems (ECCS), Oxford, UK (2006) (to appear)Google Scholar
  16. 16.
    Regnault, D., Schabanel, N., Thierry, É.: Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority (Full text). Preprint arXiv:0706.2479 [cs.DM] (2007)Google Scholar
  17. 17.
    D. Regnault, N. Schabanel, and É. Thierry. A study of stochastic 2D Minority CA: Would wearing stripes be a fatality for snob people? Research Report No ENSL-00140883, École Normale Supérieure de Lyon, 2007.Google Scholar
  18. 18.
    Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51, 123–143 (1999)CrossRefGoogle Scholar
  19. 19.
    Tarjan, R.E.: Amortized computational complexity. SIAM Journal of Algebraic and Discrete Methods 6(2), 306–318 (1985)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Damien Regnault
    • 1
    • 2
  • Nicolas Schabanel
    • 1
    • 2
  • Éric Thierry
    • 1
  1. 1.IXXI- LIP,École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07France
  2. 2.CNRS, Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120 Piso 7, Santiago de Chile 

Personalised recommendations