Characteristics of fractal cellular automata constructed from linear rules

Cellular automata (CAs) have played a significant role in the study of complex systems. Recently, the recursive estimation of neighbors (REN) algorithm was proposed to extend a CA rule with a unit rule radius to rules with larger radii. This framework enables the construction of non-uniform CAs comprising cells that follow different CA rules. A non-uniform CA, referred to as fractal CA (F-CA), which comprises fractally arranged cells, inherits certain characteristics of basic CAs, including pattern replicability and time-reversibility of linear rules. In this paper, F-CAs based on linear rules, particularly the elementary CA #90 and #150, and life-like CA B1357S1357 and B1357S02468 are investigated. Cells in the F-CAs of #90 and B1357S1357 are separated into groups by their rule radius and each group has an independent lifetime. The explicitly constructed inverse rule of F-CA of #150 is more complex than that of F-CA. The complexity of the F-CA of B1357S02468 and its inverse CA is demonstrated by image scrambling. The F-CAs can be applied to encoding and decoding processes for encryption systems.

[1]  Bastien Chopard,et al.  Cellular Automata Modeling of Physical Systems: Index , 1998 .

[2]  Yoshihiko Kayama,et al.  Fractal Arrangement for 2D Cellular Automata and Its Implementation for Outer-Totalistic Rules , 2018, ACRI.

[3]  W ReynoldsCraig.  Flocks, herds and schools: A distributed behavioral model , 1987 .

[4]  Per Brinch Hansen,et al.  Parallel cellular automata: A model program for computational science , 1993, Concurr. Pract. Exp..

[5]  G. A. Edgar.  Classics On Fractals , 2019 .

[6]  Andrew Adamatzky,et al.  Game of Life Cellular Automata , 2010 .

[7]  G B Ermentrout,et al.  Cellular automata approaches to biological modeling. , 1993, Journal of theoretical biology.

[8]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[9]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[10]  David Eppstein.  Growth and Decay in Life-Like Cellular Automata , 2010, Game of Life Cellular Automata.

[11]  Craig W. Reynolds.  Flocks, herds, and schools: a distributed behavioral model , 1998 .

[12]  Elwyn R. Berlekamp,et al.  Winning Ways for Your Mathematical Plays, Volume 2 , 2003 .

[13]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[14]  P. P. Chaudhuri,et al.  A Survey on Cellular Automata ∗ , 2003 .

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

[16]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[17]  Yoshihiko Kayama.  Expansion of Perception Area in Cellular Automata Using Recursive Algorithm , 2016 .

[18]  Computers in Eastern Europe , 1970 .

[19]  Stephen Wolfram,et al.  Theory and Applications of Cellular Automata , 1986 .

[20]  John von Neumann,et al.  Theory Of Self Reproducing Automata , 1967 .

[21]  Yoshihiko Kayama.  Cellular automata in fractal arrangement , 2018, Artificial Life and Robotics.

[22]  Yoshihiko Kayama.  Network representation of cellular automata , 2011, 2011 IEEE Symposium on Artificial Life (ALIFE).

[23]  E. Fredkin.  Digital mechanics: an informational process based on reversible universal cellular automata , 1990 .

[24]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[25]  John Briggs,et al.  Fractals : the patterns of chaos : a new aesthetic of art, science, and nature , 1992 .

[26]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[27]  Marián Boguñá,et al.  Self-similarity of complex networks and hidden metric spaces , 2007, Physical review letters.

[28]  Yoshihiko Kayama.  Extension of cellular automata via the introduction of an algorithm for the recursive estimation of neighbors , 2016, Artificial Life and Robotics.