Nilpotency and Limit Sets of Cellular Automata

Authors:
Pierre Guillon;Gaétan Richard
Affiliations:
Laboratoire d'Informatique Gaspard Monge, UMR CNRS 8049, Université Paris-Est, Marne la Vallée Cedex 2, France 77454;Laboratoire d'Informatique Fondamentale de Marseille, Aix-Marseille Université, CNRS, Marseille, France 13013
Venue:
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Year:
2008

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Abstract

A one-dimensional cellular automaton is a dynamical system which consisting in a juxtaposition of cells whose state changes over discrete time according to that of their neighbors. One of its simplest behaviors is nilpotency: all configurations of cells are mapped after a finite time into a given "null" configuration. Our main result is that nilpotency is equivalent to the condition that all configurations converge towards the null configuration for the Cantor topology, or, equivalently, that all cells of all configurations asymptotically reach a given state.