Rice's theorem for generic limit sets of cellular automata

TL;DR

This paper proves that any non-trivial property of the generic limit sets of cellular automata is undecidable, highlighting fundamental limits in understanding their asymptotic behaviors.

Contribution

It establishes a Rice-like theorem for generic limit sets of cellular automata, showing all non-trivial properties are undecidable.

Findings

All non-trivial properties of generic limit sets are undecidable.

The result extends Rice's theorem to the context of cellular automata.

Highlights the complexity of analyzing asymptotic behaviors in cellular automata.

Abstract

The generic limit set of a cellular automaton is a topologically dened set of congurations that intends to capture the asymptotic behaviours while avoiding atypical ones. It was dened by Milnor then studied by Djenaoui and Guillon rst, and by T{\"o}rm{\"a} later. They gave properties of this set related to the dynamics of the cellular automaton, and the maximal complexity of its language. In this paper, we prove that every non trivial property of these generic limit sets of cellular automata is undecidable.