Complexity of Generic Limit Sets of Cellular Automata

TL;DR

This paper investigates the structural and computational properties of generic limit sets in cellular automata, revealing complexity bounds and restrictions related to their language and periodicity.

Contribution

It establishes upper bounds on the computational complexity of generic limit sets and identifies structural restrictions, advancing understanding of cellular automata dynamics.

Findings

The language of a generic limit set can be at most a0^0_3-hard.

Structural restrictions apply to generic limit sets with a global period.

Lower complexity bounds are identified in special cases.

Abstract

The generic limit set of a topological dynamical system of the smallest closed subset of the phase space that has a comeager realm of attraction. It intuitively captures the asymptotic dynamics of almost all initial conditions. It was defined by Milnor and studied in the context of cellular automata, whose generic limit sets are subshifts, by Djenaoui and Guillon. In this article we study the structural and computational restrictions that apply to generic limit sets of cellular automata. As our main result, we show that the language of a generic limit set can be at most $\Sigma^0_3$-hard, and lower in various special cases. We also prove a structural restriction on generic limit sets with a global period.