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Tilings: simulation and universality

Published online by Cambridge University Press:  27 October 2010

GREGORY LAFITTE  and
MICHAEL WEISS
GREGORY LAFITTE
Affiliation:
Laboratoire d'Informatique Fondamentale de Marseille (LIF), CNRS – Aix-Marseille Université, 39 rue Joliot-Curie, F-13453 Marseille Cedex 13, France Email: Gregory.Lafitte@lif.univ-mrs.fr
MICHAEL WEISS
Affiliation:
Laboratoire I3S, CNRS – Université de Nice Sophia-Antipolis, Les Algorithmes, Bât. Euclide B, 2000 route des Lucioles, BP 121, F-06903 Sophia Antipolis Cedex, France Email: Michael.Weiss@unice.fr
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Abstract

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Wang tiles are unit-size squares with coloured edges. In this paper, we approach one aspect of the study of tiling computability: the quest for a universal tile set. Using a complex construction, based on Robinson's classical construction and its different modifications, we build a tile set (pronounced ayin) that almost always simulates any tile set. By way of Banach–Mazur games on tilings topological spaces, we prove that the set of -tilings that do not satisfy the universality condition is meagre in the set of -tilings.

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