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Journal Articles Journal of Computer and System Sciences Year : 2015

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

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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.
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Dates and versions

hal-00866094 , version 1 (25-09-2013)
hal-00866094 , version 2 (19-06-2015)

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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⟩
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