The difficulties in classifying cellular automata and networks based on their global behavior have been explained in Chapter 8. The results tend to indicate that a complete classification is, at least in many computational ways, impractical or unfeasible. The study of the specific properties of global behavior of arbitrary automata in terms of their local rules is indeed a most interesting and difficult problem.
Among other reasons for this difficulty is that these problems appear to be of a discrete and combinatorial nature, where classical optimization and analysis are not directly applicable. This chapter deals with positive results that shed light on the long-term behavior of global dynamics. Each technique has only been partially successful and they can be roughly classified as either discrete/combinatorial or analytic. The first section presents some results obtained through combinatorial techniques. Later, several analogies with continuous classical dynamical systems are exploited to gain insight into the nature of the long-term behavior of cellular networks. Despite complex behavior as chaotic as that of maps on the interval, many of them exhibit an interesting property of observability through simulation on computing devices under limitations such as bounded precision, rounding errors, and noise.
You can fool all the people some of the time. You can even fool some of the people all the time. But you can’t fool all the people all the time.
Abraham Lincoln
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References
R. Bartlett: Discrete computation in the continuum. Doctoral dissertation, Department of Mathematical Sciences, The University of Memphis, 1994
R. Bartlett, M. Garzon: Computation universality of monotonic maps of the interval. Preprint
R. Bartlett, M. Garzon: Monomial cellular automata. Complex Systems 7:5 (1993) 367–388
R. Bennet: Countable dense homogenous spaces, Fund. Math. 74 (1971)189–194
R. Bowen: Entropy for group endomorphisms and homogeneous spaces.Trans. Amer. Math. Soc. 153 (1971) 401–414
R. Bowen: On axiom A diffeomorphisms. In: CBMS Regional Conference Series in Math. 35, American Mathematical Society, 1978
L. Blum, M. Shub, S. Smale: On a theory of computation over the real numbers; NP-completeness, recursive functions and universal machines. Bull. AMS 21(1989) 1–46
L. Block, J. Guckenheimer, M. Misiurewicz, L.S. Young: Periodic points and topological entropy of one-dimensional maps. In: T. Nitecki and C. Robinson (eds.): Global Theory of Dynamical Systems. Lecture Notes in Mathematics 819. Springer-Verlag, New York, 1980, pp. 18–34
F. Botelho, M. Garzon: On dynamical properties of neural networks. Complex Systems 5:4 (1991) 401–413
F. Botelho, M. Garzon: Boolean neural networks are observable. Theoret. Comput. Sci 134 (1994) 51–61. Corrigendum, ibid., forthcoming
P. Collet, J.P. Eckmann. Iterated maps on the interval as dynamical systems. In: Progress in Physics, vol. 1. Birkhäuser, Boston, 1980
R. Cordovil, R. Dilao, A. Noronha Da Costa: Periodic orbits of additive cellular automata. Discrete Comput. Geometry (1986) 277–288
M. Cosnard, M. Garzon, P. Koiran: Computability properties of low- dimensional dynamical systems, Ext. Abs. in STACS’93. Lecture Notes in Computer Science, Vol. 665. Springer-Verlag, New York, 1993. Full version in Theoret. Comput. Sci. (1994), in press
E. Coven, I. Kan, J. Yorke: Pseudo-orbit shadowing in the family of tent maps. Trans. AMS 308 (1988) 227–241
P. Cull: Dynamics of neural nets. Trends in Biolog. Cybernetics 1 (1991)
R.L. Devaney: An introduction to chaotic dynamical systems. Addison-Wesley Publishing, Reading MA, 1986
M. Garzon, F. Botelho: Observability of neural network behavior. Ext. Abs. in: Proc. 6th Neural Information Processing Systems Conference, J. Cowan et al. (eds.). Morgan-Kaufmann CA, 1993 pp 455–462
R. Gilman: Classes of linear automata. Ergodic Th. Dynam. Syst. 7 (1987) 108–118
R Grassberger: Chaos and diffusion in deterministic cellular automata. Physica D 10(1984) 52–58
P. Guan, Y. He: Exact results for deterministic cellular automata with additive rules. J. Statistical Physics 43:3/4 (1978) 445–455
H.A. Gutowitz: Transients, cycles and complexity in cellular automata. Phys. Review A (1991)
G.A. Hedlund: Endomorphisms and automorphisms of the shift dynamical System. Math. Syst. Theory 3 (1969) 320–375
J.G. Hocking, G.S. Young: Topology. Addison-Wesley, Boston MA, 1969
H. Ito: Intriguing properties of global structure in some class of finite cellular automata. Physica D 31(1988) 318–338
E. Jen: Cylindric cellular automata. Comm. Math. Physics 118 (1988) 569–590
E. Jen: Linear cellular automata and recurrence systems in finite fields. Comm. Math. Physics 119(1988) 13–28
E. Jen: Exact solvability and quasi-periodicity of one-dimensional cellular automata. Nonlinearity 4 (1991) 251–276
P. Kurka: Universal computation in dynamical systems. Preprint, Charles University, Praha, Czech Republic, 1993
P. Kurka: Simulation in dynamical systems and Turing machines. Preprint, Charles University, Praha, Czech Republic, 1993
P. Kurka: Languages, equicontinuity and attractors in linear cellular automata. Preprint, Charles University, Praha, Czech Republic, 1993
T. Li, J. Yorke: Period three implies chaos. Amer. Math. Monthly 82(1975) 985–992
O. Martin, A.M. Odlyzko, S. Wolfram: Algebraic properties of cellular automata. Comm. Math. Phys. 93(1984) 219–258.
C. Moore: Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991) 199–230.
C. Moore: Generalized one-sided shifts and maps of the interval. Nonlinearity 4 (1991) 727–745
C. Moore: Smooth maps of the interval and the real line capable of universal computation. Preprint, 1993
A. Morimoto: Some stabilities of group automorphisms. Kyoto Daigaku Sur. Kokyuroku 313 (1977) 148–164
C. Preston: Iterates of piecewise monotone mappings on an interval. In: Lecture Notes in Mathematics, Vol. 1347. Springer-Verlag, New York, 1988
H. T. Siegelman, E. D. Sontag: Turing computation with neural nets Appl. Math. Lett. 4:6 (1991) 77–80
H. T. Siegelman, E. D. Sontag: On the computational power of neural nets. In: Proc. Fifth ACM Workshop on Computational Learning Theory (COLT). Morgan-Kaufmann, San Mateo CA, 1992
P. Walters: On the pseudo-orbit tracing property and its relationship to stability. Lecture Notes in Mathematics, Vol. 668. Springer-Verlag, New York, 1978, pp 231–244
S. Wolfram: Computation theory of cellular automata, Comm. Math. Physics 96 (1984) 15–57
A. Wuensche and M.J. Lesser: The global dynamics of cellular automata: an atlas of basins of attraction fields of one-dimensional cellular automata. In: Santa Fe Institute Studies in the Science of Complexity, Vol. 1. Addison-Wesley, Reading MA, 1992
A. Wuensche: Basins of attraction in disordered networks, In: I. Alexander, J. Taylor (eds.). Artificial Neural Networks. Elsevier, Amsterdam, 1992
A. Wuensche: Complexity in one-dimensional cellular automata: gliders, basins of attraction and the Z-parameter. Cognitive Science Research paper, University of Sussex, 1993
A. Wuensche: Memory far from equilibrium: basins of attraction of random boolean networks. In: Proc. Conf. on Artifical Life, Université Libre de Bruxelles, 1993
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Garzon, M. (1991). Asymptotic Behavior. In: Models of Massive Parallelism. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77905-3_9
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