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

•We characterize the set of μ-limit sets of cellular automata.•We prove that the language of these limit sets can be Σ3Σ3-complete.•We prove a Rice theorem for μ-limit sets of cellular automata.

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.