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Natural Computing

Cellular automata on group sets and the uniform Curtis–Hedlund–Lyndon theorem

  • 04-10-2017
Published in:
Author
Simon Wacker
Published in
Natural Computing | Issue 3/2019

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Abstract

We introduce cellular automata whose cell spaces are left-homogeneous spaces, show that their global transition functions are closed under composition, prove a uniform as well as a topological variant of the Curtis–Hedlund–Lyndon theorem, and deduce that such an automaton is invertible if and only if its global transition function is bijective. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations.

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previous article Cellular automata and finite groups
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Title
Cellular automata on group sets and the uniform Curtis–Hedlund–Lyndon theorem
Author
Simon Wacker
Publication date
04-10-2017
Publisher
Springer Netherlands
Published in
Natural Computing / Issue 3/2019
Print ISSN: 1567-7818
Electronic ISSN: 1572-9796
DOI
https://doi.org/10.1007/s11047-017-9645-y
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