$μ$-Limit Sets of Cellular Automata from a Computational Complexity Perspective

Abstract: This paper concerns $\mu$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $\mu$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $\mu$-limit sets can have a $\Sigma_3^0$-hard language, second, they can contain only $\alpha$-complex configurations, third, any non-trivial property concerning them is at least $\Pi_3^0$-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.