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

Laurent Boyer (1) , Martin Delacourt (2) , Victor Poupet (3) , Mathieu Sablik (4) , Guillaume Theyssier (5)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
3 ESCAPE - Systèmes complexes, automates et pavages
4 LATP - Laboratoire d'Analyse, Topologie, Probabilités
5 LAMA - Laboratoire de Mathématiques

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.

Mots clés: μ-limit sets; cellular automata; arithmetical hierarchy; Rice theorem

Domaines: Mathématique discrète [cs.DM]

Laurent Boyer, Martin Delacourt, Victor Poupet, Mathieu Sablik, Guillaume Theyssier. μ-Limit Sets of Cellular Automata from a Computational Complexity Perspective. Journal of Computer and System Sciences, 2015, 81 (8), pp.1623-1647. ⟨10.1016/j.jcss.2015.05.004⟩. ⟨hal-00866094v2⟩