A full computation-relevant topological dynamics classification of elementary cellular automata.

Cellular automata are both computational and dynamical systems. We give a complete classification of the dynamic behaviour of elementary cellular automata (ECA) in terms of fundamental dynamic system notions such as sensitivity and chaoticity. The "complex" ECA emerge to be sensitive, but not chaotic and not eventually weakly periodic. Based on this classification, we conjecture that elementary cellular automata capable of carrying out complex computations, such as needed for Turing-universality, are at the "edge of chaos."

[1]  Nicolas Ollinger.  Universalities in cellular automata a (short) survey , 2008, JAC.

[2]  Stephen Wolfram,et al.  Universality and complexity in cellular automata , 1983 .

[3]  Robert H. Gilman.  Classes of linear automata , 1987 .

[4]  A. Odlyzko,et al.  Algebraic properties of cellular automata , 1984 .

[5]  S. Wolfram.  Statistical mechanics of cellular automata , 1983 .

[6]  Ruedi Stoop,et al.  Global Dynamics of Finite Cellular Automata , 2008, ICANN.

[7]  Karel Culik,et al.  Undecidability of CA Classification Schemes , 1988, Complex Syst..

[8]  Leonid A. Bunimovich,et al.  Complexity of Dynamics as Variability of Predictability , 2004 .

[9]  P. Kurka.  Languages, equicontinuity and attractors in cellular automata , 1997, Ergodic Theory and Dynamical Systems.

[10]  G. A. Hedlund.  Endomorphisms and automorphisms of the shift dynamical system , 1969, Mathematical systems theory.

[11]  Pierre Tisseur,et al.  Some properties of cellular automata with equicontinuity points , 2000 .

[12]  Enrico Formenti,et al.  On undecidability of equicontinuity classification for cellular automata , 2003, DMCS.

[13]  Leon O. Chua,et al.  A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part I: Threshold of Complexity , 2002, Int. J. Bifurc. Chaos.

[14]  Gianpiero Cattaneo,et al.  Investigating topological chaos by elementary cellular automata dynamics , 2000, Theor. Comput. Sci..

[15]  Jean-Charles Delvenne,et al.  Decidability and Universality in Symbolic Dynamical Systems , 2004, Fundam. Informaticae.

[16]  Petr Kůrka,et al.  Topological dynamics of one-dimensional cellular automata , 2007 .

[17]  Paola Flocchini,et al.  On the Relationship Between Boolean and Fuzzy Cellular Automata , 2009, AUTOMATA.

[18]  R Stoop,et al.  Natural computation measured as a reduction of complexity. , 2004, Chaos.