Regional Control of Probabilistic Cellular Automata

  • Franco Bagnoli
  • Sara Dridi
  • Samira El Yacoubi
  • Raúl Rechtman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11115)


Probabilistic Cellular Automata are extended stochastic systems, widely used for modelling phenomena in many disciplines. The possibility of controlling their behaviour is therefore an important topic. We shall present here an approach to the problem of controlling such systems by acting only on the boundary of a target region.


Probabilistic cellular automata Control theory Boundary control Reachability 



R.S. acknowledges partial financial support from PPA-DGAPA-UNAM.


  1. 1.
    See for instance the series of proceedings of the ACRI (Cellular Automata for Research and Industry) conferences Cellular Automata (Lectures Notes in Computer Science, Springer): ACRI2002, LNCS 2493, DOI:;ACRI2004, LNCS 3305,; ACRI2006, LNCS 4173,; ACRI2008, LNCS 5191,; ACRI2010, LNCS 6350,; ACRI2012, LNCS 7495,; ACRI2014, LNCS 8751,; ACRI2016, LNCS 9863,
  2. 2.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437 (1969). Scholar
  3. 3.
    Damiani, C., Serra, R., Villani, M., Kauffman, S.A., Colacci, A.: Cell-cell interaction and diversity of emergent behaviours. IET Syst. Biol. 5, 137 (2011). Scholar
  4. 4.
    Deutsch, A., Dormann, S.: Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis. Birkhäuser, Berlin (2005). Scholar
  5. 5.
    Ermentrout, G., Edelstein-Keshet, L.: Cellular automata approaches to biological modeling. J. Theor. Biol. 160, 97–133 (1993). Scholar
  6. 6.
    Boccara, N., Goles, E., Martínez, S., Picco, P. (eds.): Cellular Automata and Cooperative Systems. Nato Science Series C, vol. 396. Springer, Amsterdam (1983). Scholar
  7. 7.
    Chopard, B., Droz, M.: Cellular Automata Modeling of Physical Systems. Cambridge University Press, Cambridge (1998). Scholar
  8. 8.
    Codd, E.F.: Cellular Automata. Academic Press, New York (1968). ISBN 0121788504zbMATHGoogle Scholar
  9. 9.
    Burks, A.W.: Essays on Cellular Automata. University of Illinois Press, Champaign (1970)zbMATHGoogle Scholar
  10. 10.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays, vol. 2. Academic Press, New York (1982). EAN 9781568811420zbMATHGoogle Scholar
  11. 11.
    Vichniac, G.: Simulating physics with cellular automata. Phys. D 10, 96–115 (1984). Scholar
  12. 12.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601 (1983). Scholar
  13. 13.
    Wolfram, S.: Universality and complexity in cellular automata. Physica 10D, 1 (1984). Scholar
  14. 14.
    Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334, 3–33 (2005). Scholar
  15. 15.
    Domany, E., Kinzel, W.: Equivalence of cellular automata to Ising models and directed percolation. Phys. Rev. Lett. 53, 311–314 (1984). Scholar
  16. 16.
    Louis, P.-Y., Nardi, F. (eds.): Probabilistic Cellular Automata, Emergence, Complexity and Computation, vol. 27. Springer, Basel (2018). Scholar
  17. 17.
    Zerrik, E., Boutoulout, A., El Jai, A.: Actuators and regional boundary controllability for parabolic systems. Int. J. Syst. Sci. 31, 73–82 (2000). Scholar
  18. 18.
    Lions, J.: Controlabilité exacte des systèmes distribueés. CRAS, Série I(302), 471–475 (1986)zbMATHGoogle Scholar
  19. 19.
    Lions, J.: Exact controllability for distributed systems. Some trends and some problems. In: Spigler, R. (ed.) Applied and Industrial Mathematics. MAIA, vol. 56, pp. 59–84. Springer, Dordrecht (1991). Scholar
  20. 20.
    Russell, D.: Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20, 639–739 (1978). Scholar
  21. 21.
    El Yacoubi, S., El Jai, A., Ammor, N.: Regional controllability with cellular automata models. In: Bandini, S., Chopard, B., Tomassini, M. (eds.) ACRI 2002. LNCS, vol. 2493, pp. 357–367. Springer, Heidelberg (2002). Scholar
  22. 22.
    Fekih, A.B., El Jai, A.: Regional Analysis of a Class of Cellular Automata Models. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 48–57. Springer, Heidelberg (2006). Scholar
  23. 23.
    El Yacoubi, S.: Mathematical method for control problems on cellular automata models. Int. J. Syst. Sci. 39(5), 529–538 (2008). Scholar
  24. 24.
    Bagnoli, F., El Yacoubi, S., Rechtman, R.: Synchronization and control of cellular automata. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) ACRI 2010. LNCS, vol. 6350, pp. 188–197. Springer, Heidelberg (2010). Scholar
  25. 25.
    Bagnoli, F., Rechtman, R., El Yacoubi, S.: Control of cellular automata. Phys. Rev. E 86, 066201 (2012). Scholar
  26. 26.
    Bagnoli, F., El Yacoubi, S., Rechtman, R.: Toward a boundary regional control problem for Boolean cellular automata. Nat. Comput. (2017).
  27. 27.
    Bagnoli, F., El Yacoubi, S., Rechtman, R.: Control of cellular automata. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science. Springer, Heidelberg (2018). Scholar
  28. 28.
    Bagnoli, F., Rechtman, R.: Regional synchronization of a probabilistic cellular automaton. In: Mauri, G., et al. (eds.) ACRI 2018, LNCS, vol. 11115. pp. 255–263. Springer, Heidelberg (2018)Google Scholar
  29. 29.
    Bagnoli, F.: Cellular automata in dynamical modelling in biotechnologies. In: Bagnoli, F., Lió, P., Ruffo, S. (eds.) p. 3. World Scientific, Singapore, (1998).
  30. 30.
    Bagnoli, F., Boccara, B., Rechtman, R.: Nature of phase transitions in a probabilistic cellular automaton with two absorbing states. Phys. Rev. E 63, 046116 (2001). Scholar
  31. 31.
    Vichniac, G.: Boolean derivatives on cellular automata. Physica 10D, 96 (1984). Scholar
  32. 32.
    Bagnoli, F.: Boolean derivatives and computation of cellular automata. Int. J. Mod. Phys. C. 3, 307 (1992). Scholar
  33. 33.
    Bagnoli, F., Rechtman, R.: Synchronization and maximum Lyapunov exponents of cellular automata. Phys. Rev. E 59, R1307 (1999). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Franco Bagnoli
    • 1
    • 2
  • Sara Dridi
    • 3
  • Samira El Yacoubi
    • 3
  • Raúl Rechtman
    • 4
  1. 1.Department of Physics and Astronomy and CSDCUniversity of FlorenceSesto FiorentinoItaly
  2. 2.INFN, sez. FirenzeSesto FiorentinoItaly
  3. 3.Team Project IMAGES_ESPACE-Dev, UMR 228 Espace-Dev IRD UA UM UG UR, University of Perpignan Via DomitiaPerpignan CedexFrance
  4. 4.Instituto de Energías RenovablesUniversidad Nacional Autónoma de MéxicoTemixcoMexico

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