On the complexity of limit sets of cellular automata associated with probability measures

Authors:
Laurent Boyer;Victor Poupet;Guillaume Theyssier
Affiliations:
LIP (UMR 5668 — CNRS, ENS Lyon, UCB Lyon, INRIA), ENS Lyon, LYON cedex 07, France;LIP (UMR 5668 — CNRS, ENS Lyon, UCB Lyon, INRIA), ENS Lyon, LYON cedex 07, France;LAMA (UMR 5127 — CNRS, Université de Savoie), Université de Savoie, Le Bourget-du-lac cedex, France
Venue:
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Year:
2006

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Abstract

We study the notion of limit sets of cellular automata associated with probability measures (μ-limit sets). This notion was introduced by P. Kůrka and A. Maass in [1]. It is a refinement of the classical notion of ω-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterization of the persistent language for non sensitive cellular automata associated with Bernoulli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their μ-limit set) is neither recursively enumerable nor co-recursively enumerable.