Abstract

Cellular automata and discrete neural networks constitute very simple and general models that seem to capture the fundamental features of a variety of highly complex systems. Their study offers the possibility of obtaining some understanding of the most important and characteristic properties of complex and self-organizing systems, the evolution of which currently appears chaotic, disorganized and beyond the scope of known laws of nature.

Keywords

Entropy Conglomerate Incompressibility Wolfram Rovan 

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References

  1. [A-K]
    A. Akira, M. Kimura: Decomposition phenomenon in one-dimensional scope- three tesselation automata with arbitrary number of states. Inf. and Control 34 (1977) 296–313.MATHCrossRefGoogle Scholar
  2. [A-K2]
    A. Akira, M. Kimura: Completeness in tesselation automata. Inf. and Control 35(1977) 52–86MATHCrossRefGoogle Scholar
  3. [B]
    R. Bowen: Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971) 401–414MathSciNetMATHCrossRefGoogle Scholar
  4. [C-Y]
    K. Culik, S. Yu: Undecidability of CA classification schemes. Complex Systems 2 (1988) 177–190MathSciNetMATHGoogle Scholar
  5. [F]
    C. Fields: Consequences of nonclassical measurement for the algorithmic description of continuous dynamical systems. J. Expt. Theor. Artif. Intell. 1 (1989) 171–178CrossRefGoogle Scholar
  6. [G-J]
    M.R. Garey, D.S. Johnson: Computers and intractabillity: a guide to the theory of NP-completeness. W.H. Freeman, San Francisco, 1978Google Scholar
  7. [Ga-Jo]
    M. Garzon, A. Jagota: Efficient neural network isomorphism testing. In: Proc. 2nd Swedish Conference on Connectionism. Erlbaum Publishers, 1994.Google Scholar
  8. [G-Z]
    M. Garzon, M. Zhang: Classifying neural networks. R.E. Trahan, Jr. (ed.). Proc. IEEE Southeast Conference, New Orleans, 1990, pp. 567–571Google Scholar
  9. [Gil]
    R. Gilman: Classes of linear automata. Ergodic Theo. and Dynam. Syst. 7 (1987) 108–118MathSciNetGoogle Scholar
  10. [Gi2]
    R. Gilman: Periodic behavior of linear automata. In: Dynamical systems. Lecture Notes in Mathematics, Vol. 1342. Springer-Verlag, Berlin, 1988, pp. 216–219Google Scholar
  11. [G-M]
    E. Goles, A. Maass, S. Martinez: On the limit set of some universal cellular automata. Theoret. Comput. Sci. 110:1 (1993) 53–78MathSciNetMATHCrossRefGoogle Scholar
  12. [Gutl]
    H. Gutowitz (ed.): Cellular automata: theory and applications. Proc. 3rd. Int. Conf. Cellular Automata, Los Alamos, 1991. Physica D 45 (1990).Google Scholar
  13. [Gut2]
    H. Gutowitz: Mean Field vs. Wolfram Classification, preprint. CNLS, Los Alamos National Lab, NM 1988Google Scholar
  14. [Gut3]
    H. Gutowitz: A hierarchical classification of cellular automata. In: [Gutl] pp. 136–158Google Scholar
  15. [Hul]
    L. Hurd: Formal language characterizations of cellular automaton limit sets. Complex Systems 1:1 (1987) 69–80MathSciNetMATHGoogle Scholar
  16. [Hu2]
    L. Hurd: The application of formal language theory to the dynamical behavior of cellular automata. Dissertation, Princeton University, 1988Google Scholar
  17. [Hu3]
    L. Hurd: The nonwandering set of a CA map. Complex Systems 2:5 (1988) 549–554MathSciNetMATHGoogle Scholar
  18. [H-K-C]
    L. Hurd, J. Kari, K. Culik: The topological entropy is uncomputable. Ergodic Theory and Dyn. Syst. 12:2 (1992) 255–265MathSciNetMATHGoogle Scholar
  19. [Hur]
    M. Hurley: Attractors in cellular automata. Ergodic Theo, and Dynam. Syst. 10 (1990) 131–140MathSciNetMATHGoogle Scholar
  20. [I]
    S. Ishii: Measure-theoretic approach to the classification of cellular automata. Disc. Appl. Math 39 (1992) 125–136MATHGoogle Scholar
  21. [Kai]
    J. Kari, The nilpotency problem of one-dimensional cellular automata. SI AMJ. Comput. 21 (1992) 571–586MathSciNetMATHGoogle Scholar
  22. [Ka2]
    J. Kari, Rice’s Theorem for the limit sets of cellular automata. Theoret. Comput. Sci. 127:2 (1994) 229–254MathSciNetMATHCrossRefGoogle Scholar
  23. [Ku3]
    P. Kürka: A comparison of finite and cellular automata. In: Math. Foundations of Computer Science MFCS, I. Privara, B. Rovan, eds. Lecture Notes in Computer Science 841, Springer-Verlag, Berlin, 1994, pp. 484–493Google Scholar
  24. [Ku4]
    P. Kürka: Languages, equicontinuity and attractors in linear cellular automata. Preprint, Charles University, Praha, Czech Republic, 1994.Google Scholar
  25. [L]
    M. Langton: Artificial life. MIT Press, Cambridge MA, 1989.Google Scholar
  26. [L-P]
    W. Li, N. Packard: The structure of elementary cellular automata rule space. Complex Systems 4 (1990) 281–297MathSciNetGoogle Scholar
  27. [L-V]
    M. Li, P. Vitänyi: An introduction to Kolmogorov complexity and its applications. Springer-Verlag, New York, 1993MATHGoogle Scholar
  28. [L-P-L]
    W. Li, N. Packard, C. Langton: Transition phenomena in cellular automata rule space. In: [Gutl], pp. 77–94Google Scholar
  29. [L-M]
    D. Lind, B. Marcus: An introduction to symbolic dynamics. Manuscript, 1993Google Scholar
  30. [Mi2]
    J. Milnor: Directional entropies of cellular automaton maps. In: Disordered systems and biological organization, Ev. Bienenstock et al., eds. Springer- Verlag, New York, 1986, pp. 113–115CrossRefGoogle Scholar
  31. [Mi3]
    J. Milnor: On the entropy geometry of cellular automata. Complex Systems 2:3 (1988) 357–386MathSciNetMATHGoogle Scholar
  32. [Mu]
    S. Muroga, I. Toda, S. Takasu: Theory of majority decision elements, J. Franklin Inst. 271 (1961) 376–418MathSciNetMATHCrossRefGoogle Scholar
  33. [P-S]
    I. Parberry, G. Schnitger: Parallel computation with threshold functions. J. Comput. Syst. Sci. 36 (1988) 278–302MathSciNetMATHCrossRefGoogle Scholar
  34. [Svo]
    K. Svozil: Constructive chaos by cellular automata and possible sources of an arrow of time. In: Proc. 3rd Int. Conf. in Cellular Automata, 1989. Physica D 45 (1990) 420–427Google Scholar
  35. [Su]
    K. Sutner: A note on Culik-Yu classes. Ccpmplex Systems 3 (1989) 107–115MathSciNetMATHGoogle Scholar
  36. [Wol]
    S. Wolfram: Twenty problems in the theory of cellular automata. Physica Scripta T9 (1985) 170–183MathSciNetMATHCrossRefGoogle Scholar
  37. [Wo2]
    S. Wolfram: Statistical mechanics of cellular automata. Rev. of Modern Phys. 55:3 (1983) 601–644MathSciNetMATHCrossRefGoogle Scholar
  38. [Wo3]
    S. Wolfram: Random sequence generation by cellular automata. Adv. in Applied Math. 7 (1986) 123–169. Reprinted in [Wo4]MathSciNetMATHCrossRefGoogle Scholar
  39. [Wo4]
    S. Wolfram: Theory and Applications of cellular automata. World Scientific Publishing, Singapore, 1986.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Max Garzon
    • 1
  1. 1.Department of Mathematical SciencesThe University of MemphisMemphisUSA

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