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Characteristics of fractal cellular automata constructed from linear rules

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Abstract

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.

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Acknowledgements

The author wishes to thank all the anonymous reviewers for valuable comments and suggestions. This research was supported in part by a grant from BAIKA Gakuen, Japan.

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  1. Department of Media and Information, BAIKA Women’s University, 2-19-5 Shukuno-sho, Ibaraki, Osaka, 567-8578, Japan

    Yoshihiko Kayama

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Correspondence to Yoshihiko Kayama.

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Kayama, Y. Characteristics of fractal cellular automata constructed from linear rules. Artif Life Robotics 25, 8–16 (2020). https://doi.org/10.1007/s10015-019-00567-5

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