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

Highlights

• We characterize the set of μ-limit sets of cellular automata.

• We prove that the language of these limit sets can be ${\mathrm{\Sigma }}_3$-complete.

• We prove a Rice theorem for μ-limit sets of cellular automata.

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 ${\mathrm{\Sigma }}_3^0$-hard language, second, they can contain only α-complex configurations, third, any non-trivial property concerning them is at least ${\mathrm{\Pi }}_3^0$-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.