On the undecidability of the limit behavior of Cellular Automata

Abstract

Cellular Automata (CA) are discrete dynamical systems and an abstract model of parallel computation. The limit set of a cellular automaton is its maximal topological attractor. A well-known result, due to Kari, says that all nontrivial properties of limit sets are undecidable. In this paper we consider the properties of limit set dynamics, i.e. properties of the dynamics of CA restricted to their limit sets. There can be no equivalent of Kari’s theorem for limit set dynamics. Anyway we show that there is a large class of undecidable properties of limit set dynamics, namely all properties of limit set dynamics which imply stability or the existence of a unique subshift attractor. As a consequence we have that it is undecidable whether the cellular automaton map restricted to the limit set is the identity map and whether it is closing, injective, expansive, positively expansive and transitive.