Undecidable Properties of Limit Set Dynamics of Cellular Automata

TL;DR

This paper investigates the undecidability of properties related to the dynamics of cellular automata on their limit sets, extending known results about the undecidability of limit set properties.

Contribution

It introduces a broad class of undecidable properties of limit set dynamics, including stability and the existence of a unique subshift attractor, which generalizes previous undecidability results.

Findings

Undecidability of whether the CA map on the limit set is the identity

Undecidability of properties like closing, injective, expansive, and transitive dynamics

No equivalent of Kari's theorem for limit set dynamics

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 know result, due to Kari, says that all nontrivial properties of limit sets are undecidable. In this paper we consider properties of limit set dynamics, i.e. properties of the dynamics of Cellular Automata 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, closing, injective, expansive, positively expansive, transitive.