Structural and computational complexity of tilings and cellular automata

Abstract : This thesis examines the complexity of tilings and cellular automata. The analysis begins by structural considerations: quasiperiodic tilings. To any set of tiles that tiles the plane, we associate a quasiperiodicity function that quantifies complexity. Firstly, it is shown that any "reasonable" function may be captured by a set of tiles and that there are tilings whose quasiperiodicity function grows faster than any computable function. Then we prove a Rice theorem for tilings: the set of all tile sets that recognize the same tilings as another tile set is recursively enumerable and undecidable. Finally, we transpose our results in the context of cellular automata. The second part of our work concerns the study of cellular automata in terms of dynamical systems, particularly chaotic controllers. The usual definitions classifying chaotic controllers are not satisfactory. To overcome this problem, we use two new topologies. The first is called Besicovitch and removes the dominance of the central pattern in the study of the evolution of the automaton. It brings several results, the first indicating that our new workspace is acceptable to the study of cellular automata as dynamical systems; the latter shows that the notion of chaos remains, through the definition of sensitivity to initial conditions, but the most chaotic classes are empty. The second topology employed is defined using algorithmic complexity. The purpose of this approach is to have a distance that reflects the ease calculating a member from the other. Our results complement the earlier results. They attest formally that cellular automata can not continuously change the information in a configuration, and especially that they are incapable to create information.
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Contributor : Julien Cervelle <>
Submitted on : Wednesday, October 14, 2015 - 3:31:25 PM
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Julien Cervelle. Structural and computational complexity of tilings and cellular automata. Computer Science [cs]. Université de provence, 2002. English. ⟨tel-01208375⟩

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