An Example of Computation of the Density of Ones in Probabilistic Cellular Automata by Direct Recursion

  • Henryk FukśEmail author
Part of the Emergence, Complexity and Computation book series (ECC, volume 27)


We present a method for computing probability of occurrence of ones in a configuration obtained by iteration of a probabilistic cellular automata (PCA), starting from a random initial configuration. If the PCA is sufficiently simple, one can construct a set of words (or blocks of symbols) which is complete, meaning that probabilities of occurrence of words from this set can be expressed as linear combinations of probabilities of occurrence of these words at the previous time step. One can then set up and solve a recursion for block probabilities. We demonstrate an example of such PCA, which can be viewed as a simple model of diffusion of information or spread of rumours. Expressions for the density of ones are obtained for this rule using the proposed method.



The author acknowledges partial financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of Discovery Grant. Some calculations on which this work is based were made possible by the facilities of the Shared Hierarchical Academic Research Computing Network ( and Compute/Calcul Canada. The author thanks anonymous referees for suggestions leading to improvement of the article, including a simpler derivation of the cluster expansion formula.


  1. 1.
    Boccara, N., Fukś, H.: Modeling diffusion of innovations with probabilistic cellular automata. In: Delorme, M., Mazoyer, J. (eds.) Cellular Automata: A Parallel Model. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  2. 2.
    Cull, P., Flahive, M., Robson, R.: Difference Equations. Springer, Berlin (2004)zbMATHGoogle Scholar
  3. 3.
    Fukś, H.: Dynamics of the cellular automaton rule 142. Complex Syst. 16, 123–138 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fukś, H.: Probabilistic initial value problem for cellular automaton rule 172. DMTCS Proc. AL, 31–44 (2010)Google Scholar
  5. 5.
    Fukś, H.: Construction of local structure maps for cellular automata. J. Cell. Autom. 7, 455–488 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fukś, H., Gómez Soto, J.-M.: Exponential convergence to equilibrium in cellular automata asymptotically emulating identity. Complex Syst. 23, 1–26 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fukś, H., Skelton, A.: Response curves for cellular automata in one and two dimensions - an example of rigorous calculations. Int. J. Nat. Comput. Res. 1, 85–99 (2010)CrossRefGoogle Scholar
  8. 8.
    Fukś, H., Skelton, A.: Orbits of Bernoulli measure in asynchronous cellular automata. Discret. Math. Theor. Comput. Sci. AP, 95–112 (2011)zbMATHGoogle Scholar
  9. 9.
    Kůrka, P.: On the measure attractor of a cellular automaton. Discret. Contin. Dyn. Syst. 524–535 (2005)Google Scholar
  10. 10.
    Kůrka, P., Maass, A.: Limit sets of cellular automata associated to probability measures. J. Stat. Phys. 100, 1031–1047 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pivato, M.: Conservation laws in cellular automata. Nonlinearity 15(6), 1781 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor and Francis, London (1994)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBrock UniversitySt. CatharinesCanada

Personalised recommendations