Control of Fuzzy Cellular Automata: The Case of Rule 90

  • Samira El Yacoubi
  • Angelo B. Mingarelli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4671)


This paper is dedicated to the study of fuzzy rule 90 in relation with control theory. The dynamics and global evolution of fuzzy rules have been recently investigated and some interesting results have been obtained in [10,15,16]. The long term evolution of all 256 one-dimensional fuzzy cellular automata (FCA) has been determined using an analytical approach. We are interested in this paper in the FCA state at a given time and ask whether it can coincide with a desired state by controlling only the initial condition. We investigate two initial states consisting of a single control value u on a background of zeros and one seed adjacent to the controlled site in a background of zeros.

Keywords and Phrases

control fuzzy cellular automata Rule 90 


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  1. 1.
    Bunimovich, L.A.: Coupled Map Lattices: one Step Forward and two Steps Back. Physica D 86, 248–255 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Callier, F.M., Desoer, C.A.: Linear System Theory. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  3. 3.
    Cattaneo, G., Flocchini, P., Mauri, G., Santoro, N.: Cellular automata in fuzzy backgrounds. Physica D 105, 105–120 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Culik II, K., Yu, S.: Undecidability of CA classification schemes. Complex Systems 2, 177–190 (1988)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Curtain, R.F., Zwart, H.: An introduction to Infinite-dimensional linear systems theory. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  6. 6.
    El Yacoubi, S., El Jai, A.: Notes on control and observation in Cellular automata models. WSEAS Transaction on Computers 2(4), 1086–1092 (2003)Google Scholar
  7. 7.
    El Yacoubi, S., El Ja, 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)CrossRefGoogle Scholar
  8. 8.
    El Yacoubi, S., El Jai, A.: Cellular Automata and Spreadability. Journal of Mathematical and Computer Modelling 36, 1059–1074 (2002)zbMATHCrossRefGoogle Scholar
  9. 9.
    El Yacoubi, S., Jacewicz, P.: A genetic programming approach to structural identification of cellular automata models. Journal Of Cellular Automata (to appear)Google Scholar
  10. 10.
    Flocchini, P., Geurts, F., Mingarelli, A., Santoro, N.: Convergence and aperiodicity in fuzzy cellular automata: revisiting rule 90. Physica D 42, 20–28 (2000)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Flocchini, P., Santoro, N.: The chaotic evolution of information in the interaction between knowledge and uncertainty. In: Stonier, R.J., Yu, X.H. (eds.) Complex Systems: Mechanism of Adaptation, pp. 337–343. IOS Press, Amsterdam (1994)Google Scholar
  12. 12.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations- Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000)Google Scholar
  13. 13.
    Lee, K.Y., Chow, S., Barr, R.O.: On the control of discrete-time distributed parameter systems. Siam J. Control 10(2) (1972)Google Scholar
  14. 14.
    Lions, J.L.: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod et Gauthier- Villars, Paris (1968)Google Scholar
  15. 15.
    Mingarelli, A.B.: The global evolution of general fuzzy cellular automata. J. Cellular Automata 1(2), 141–164 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Mingarelli, A.B., El Yacoubi, S.: On the decidability of the evolution of the fuzzy cellular automata, FCA 184. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J.J. (eds.) ICCS 2006. LNCS, vol. 3993, pp. 360–366. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1966)Google Scholar
  18. 18.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, NewYork (1998)zbMATHGoogle Scholar
  19. 19.
    Wolfram, S.: Cellular Automata and Complexity. Collected Papers. World Scientific, Singapore (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Samira El Yacoubi
    • 1
  • Angelo B. Mingarelli
    • 2
  1. 1.MEPS/ASD - University of Perpignan, 52, Paul Alduy Avenue, 66860 Perpignan, CedexFrance
  2. 2.School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6Canada

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