Cellular automata over generalized Cayley graphs

Pablo Arrighi 1 S. Martiel 2 V. Nesme
1 CANA - Calcul Naturel
LIS - Laboratoire d'Informatique et Systèmes
Abstract : 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 a point; and the fact that they have a well-defined notion of translation. We propose a notion of graph associated to a language, which conserves or generalizes these features. Whereas Cayley graphs are very regular; associated graphs are arbitrary, although of a bounded degree. Moreover, it is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions for a distance on the set of configurations that makes it a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Similarly, we extend their definition to these arbitrary, bounded degree, time-varying graphs. The obtained notion of Cellular Automata over generalized Cayley graphs is stable under composition and under inversion.
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Pablo Arrighi, S. Martiel, V. Nesme. Cellular automata over generalized Cayley graphs. Mathematical Structures in Computer Science, Cambridge University Press (CUP), 2018, 18, pp.340-383. ⟨hal-01785458⟩

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