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Linear cellular automata on Cayley graphs

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Abstract

Linear cellular automata on Cayley graphs of some class of groups are studied. The injectivity and surjectivity of parallel maps are shown to be determined by their local maps. The main theorems are non-Euclidean extensions of Itô, Ôsato, and Nasu’s results on the injectivity and surjectivity of linear cellular automata [Linear Cellular Automata over Z m, Journal of Computer and System Sciences,27 (1983), pp. 125–140]. The proofs are based on Machì and Mignosi’s Garden of Eden theorem [Garden of Eden configurations for cellular automata on Cayley graphs of groups, SIAM Journal on Discrete Mathematics, 6 (1993), pp. 44–56] and properties of unique product groups. Examples of groups that allow the Itô-Ôsato-Nasu type theorem and of groups that do not are given.

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  1. Faculty of Computer and Information Sciences, Hosei University, 3- 7- 2 Kajino-cho, 184-8584, Koganeishi, Japan

    Shûichi Yukita

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  1. Shûichi Yukita
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Yukita, S. Linear cellular automata on Cayley graphs. Japan J. Indust. Appl. Math. 18, 15–24 (2001). https://doi.org/10.1007/BF03167352

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  • DOI: https://doi.org/10.1007/BF03167352

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