Inducing an Order on Cellular Automata by a Grouping Operation

A grouped instance of a cellular automaton (CA) is another one obtained by grouping several states into blocks and by letting interact neighbor blocks. Based on this operation (and on the subautomaton notion), a preorder on the set of one dimensional CA is introduced. It is shown that (CA,) admits a global minimum and that on the bottom of (CA,) very natural equivalence classes are located. These classes remind us the rst two well-known Wolfram ones because they capture global (or dynamical) properties as nilpotency or periodicity. Non trivial properties as the undecidability of and the existence of bounded innnite chains are also proved. Finally, it is shown that (CA,) admits no maximum. This result allows us to conclude that, in a \grouping sense", there is no universal CA. Une instance groupe e d'un automate cellulaire (AC) est un autre obtenu par groupage de plusieurs etats en blocs et par l'interaction \naturelle" de ces blocs. Bas e sur cette op eration (et sur la notion de sous-automate), un pr eordre sur l'ensamble des AC a une dimension est introduit. Il est montr e que (AC,) admet un minimum global et que des classes d'equivalence tr es naturelles se trouvent en bas de (AC,). On retrouve dans ces classes les deux premi eres bien connues de Wolfram qui capturent des propiet es globales (ou dynamiques) comme la nilpotence et la p eriodicit e. Des propiet es non triviales comme l'ind ecidabilit e de et l'existence de cha ^ ines innnies born ees sont aussi prouv ees. Finalement il est montr e que (AC,) n'admet pas de maximum. Cet resultat nous permet de conclure qu'il n'existe pas de AC universel d'un point de vue \groupage". intrins eque en temps r eel. Abstract A grouped instance of a cellular automaton (CA) is another one obtained by grouping several states into blocks and by letting interact neighbor blocks. Based on this operation (and on the subautomaton notion), a preorder on the set of one dimensional CA is introduced. It is shown that (CA,) admits a global minimum and that on the bottom of (CA,) very natural equivalence classes are located. These classes remind us the rst two well-known Wolfram ones because they capture global (or dynamical) properties as nilpotency or periodicity. Non trivial properties as the undecidability of and the existence of bounded innnite chains are also proved. Finally, it is shown that (CA,) …