Global fixed point attractors of circular cellular automata and periodic tilings of the plane: Undecidability results Original Research Article

A great amount of work has been deveted to the understanding of the long-time behavior of cellular automata (CA). As for any other kind of dynamical system, the long-time behavior of a CA is described by its attractors. In this context, it has been proved that it is undecidable whether every circular configuration of a given CA evolves to some fixed point (not unique). In this paper we prove that it remains undecidable whether every circular configuration of a given CA evolves to the same fixed point. Our proof is based on properties concerning NW-deterministic periodic tilings of the plane. As a corollary it is concluded the (already proved) undecidability of the periodic tiling problem. Nevertheless, our approach could also be used to prove this result in a direct and very simple way.