Cellular Automata, Dynamics and Complexity

  • E. Goles
Part of the Springer Proceedings in Physics book series (SPPHY, volume 46)


In this paper we study some dynamical aspects of Cellular Automata. Essentially we characterize the steady state behavior for a class of local rules in the context of Potts and Bounded Threshold Automata. For Potts automata we exhibit a class that has a complex dynamics, i.e. the automaton simulates any logical function by coding binary information as gliders in a one-dimensional cellular space. On the other hand, we characterize another class which has a simple dynamical behavior: fixed points or two-cycles in the steady state. In the context of Bounded Threshold Automata with arbitrary interactions (not necessarily symmetric) we characterize its dynamics for one-dimensional cellular arrays: the only admissible cycles have period T ≤ 4. Furthermore, we give sufficient conditions to obtain a period-2 behavior in high dimensional lattices.


Cellular Automaton Local Function Local Rule Steady State Behavior Cellular Space 
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  1. [1]
    Bienenstock E., Fogelman-Soulie F., Weisbuch G., Disordered Systems and Biological Organization, Proc. Les Houches, NATO ASI series F vol.20 (Springer-Verlag, 1986).Google Scholar
  2. [2]
    Demongeot J., Goles E., Tchuente M., Cellular Automata and Dynamical Systems, (Academic Press, 1985)MATHGoogle Scholar
  3. [3]
    Fogelman-Soulié F., Robert Y., Tchuente M., Automata Networks in Computer Science, (Manchester Univ. Press, 1987).Google Scholar
  4. [4]
    Fogelman-Soulié F., Goles E., Weisbuch G., Disc. Appl. Maths. 6(1983)95.CrossRefGoogle Scholar
  5. [5]
    Fogelman-Soulié F., Goles E., Martinez S., Mejia C., “Energy functionals in Neural Networks with continuous local functions”, Res.Rep.EHEI, Paris-V, 1988, submitted to Complex Systems.Google Scholar
  6. [6]
    Goles E., Martinez S., Dynamics on Generalized Neural Networks, to appear.Google Scholar
  7. [7]
    Goles E., Olivos J., Disc. Appl. Maths.3(1981)93.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Goles E., SIAM J. on Disc, and Alg. Meths. 4(1982)529.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Goles E., Theor. Comp.Sci.41(1985)19.MATHCrossRefGoogle Scholar
  10. [10]
    Goles E., Disc. Applied Maths. 13(1986)97.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Goles E., Martinez S., Disc. Appl. Maths. 18(1987)39.MATHCrossRefGoogle Scholar
  12. [12]
    Goles E., Martinez S., “Lyapunov functionals for Automata Networks defined by cyclically monotone functions”, Res. Rep., Dep. Mat., U. de Chile, 1987, submitted to SIAM J. Disc. Maths.Google Scholar
  13. [13]
    Goles E., Vichniac G., “Lyapunov functions for parallel neural networks”, in Neural Networks for Computing, Snowbird 1986, Denker ed., Am.Inst.Phys. 151(1986)165.Google Scholar
  14. [14]
    Goles E., Vichniac G., “Attractors in synchronous Networks of multibit Automata”, Res.Rep., MIT Plasma Fusion Center, 1988, submitted to J. of Physics A.Google Scholar
  15. [15]
    Goles E., Fogelman-Soulie F., Pellegrin D., Disc. Applied Maths. 12(1985)261.MATHCrossRefGoogle Scholar
  16. [16]
    Goles E., Olivos J., Inf. and Control51(1981)2.Google Scholar
  17. [17]
    Hopfield J. J., Proc. Nat. Acad. Sc. USA 79(1982)2554.MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    Hopfield J. J., Tank D. W., Biol.Cybern. 52(1985)141.MathSciNetMATHGoogle Scholar
  19. [19]
    Shingai R., Inf. and Control 41(1979).Google Scholar
  20. [20]
    Smith A. R., J.Assoc. Computing Machinery, 1(1971)339.Google Scholar
  21. [21]
    Ulam S., “Some mathematical problems connected with patterns of growth figures”, inEssays on Cellular Automata, Burks A. W. ed., (Univ. Illinois Press, 1970).Google Scholar
  22. [22]
    Von Neumann J., Theory of self-reproducing automata, (Univ. of Illinois Press, Urbana, 1966).Google Scholar
  23. [23]
    Wolfram S., Theory and Applications of Cellular Automata, (World Scientific, 1986).MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • E. Goles
    • 1
  1. 1.Departamento de Matematicas, Escuela de IngenieriaU. de ChileSantiagoChile

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