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Cellular automata over generalized Cayley graphs

Published online by Cambridge University Press:  29 May 2017

PABLO ARRIGHI ,
SIMON MARTIEL  and
VINCENT NESME
PABLO ARRIGHI
Affiliation:
Aix-Marseille Univ., CNRS, LIF, Marseille and IXXI, Lyon, France Email: pablo.arrighi@univ-amu.fr
SIMON MARTIEL
Affiliation:
INRIA Saclay, ENS Cachan, LSV, 61 avenue du président Wilson 94235 Cachan Email: martiel@lsv.ens-cachan.fr
VINCENT NESME
Affiliation:
Université de Grenoble, LIG, 220 rue de la chimie, 38400 Saint-Martin-d'Hères, France Email: vnesme@imag.fr
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Abstract

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It is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions over a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that they have a well-defined notion of translation. We propose a notion of graphs, which preserves or generalizes these features. Whereas Cayley graphs are very regular, generalized Cayley graphs are arbitrary, although of a bounded degree. We extend cellular automata theory to these arbitrary, bounded degree, time-varying graphs. The obtained notion of cellular automata is stable under composition and under inversion.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Special Issue 3: Mind, Mechanism and Mathematics: Computability Unchained , March 2018 , pp. 340 - 383
Copyright
Copyright © Cambridge University Press 2017 

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