μ-Limit sets of cellular automata from a computational complexity perspective

Abstract

This paper concerns μ-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial μ-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, μ-limit sets can have a Σ30-hard language, second, they can contain only α-complex configurations, third, any non-trivial property concerning them is at least Π30-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.