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Automates cellulaires : un modèle de complexités

Abstract : We study the model of cellular automata through two complementary aspects: local syntactic representations and global dynamics. We aim at establishing new links between these aspects using a set of approaches ranging from combinatorics to algebraic tools and computability theory. First, while studying local structures of transition rules, we introduce a new class of cellular automata (namely captive cellular automata) which are defined by an elementary local constraint. We establish a 0-1 law over that class and deduce that almost all captive cellular automata are intrinsically universal. However, we show that it is undecidable to determine whether a captive cellular automaton is intrinsically universal or not. In a second part, we focus on global properties of cellular automata while trying to disregard their syntactic representations. Our matter is then to study classifications and notions of complexity according to that global point of view. The key tool is the notion of simulation. We extend previous results from N. Ollinger concerning simulation pre-orders (with new relations of simulation and new properties inducing ideal or filter structures). We also study the Cartesian product over such structures. We establish a construction which can be considered as a limit of Cartesian products and allows us to exhibit infinite increasing chains of length omega+omega in one of the studied pre-orders. Finally, we focus on sequential dynamics and Turing-universal cellular automata. We construct an infinite lattice of Turing-universal cellular automata which are all infinitely far away from any intrinsically universal cellular automaton.
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Contributor : Guillaume Theyssier <>
Submitted on : Thursday, August 2, 2007 - 5:05:02 PM
Last modification on : Friday, November 6, 2020 - 3:28:48 AM
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  • HAL Id : tel-00166295, version 1



Guillaume Theyssier. Automates cellulaires : un modèle de complexités. Mathématiques [math]. Ecole normale supérieure de lyon - ENS LYON, 2005. Français. ⟨tel-00166295⟩



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