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Complex tilings

Published online by Cambridge University Press:  12 March 2014

Bruno Durand ,
Leonid A. Levin  and
Alexander Shen
Bruno Durand
Affiliation:
Laboratoire D'Informatique Fondamentale De Marseille, 39, Rue Joliot-Curie, 13453 Marseille, France, E-mail: Bruno.Durand@lif.univ-mrs.fr
Leonid A. Levin
Affiliation:
Boston University, Computer Science Department, 111 Cummington St., Boston. MA 02215, USA, E-mail: Lnd@bu.edu
Alexander Shen
Affiliation:
Laboratoire D'Informatique Fondamentale De Marseille, and, Institute of Problems of Information Transmission, Moscow., Russia, E-mail: alexander.shen@lif.univ-mrs.fr, E-mail: shen@mccme.ru
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Abstract

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings—that we also develop in terms of Turing degrees of unsolvability.

Keywords

TilingsKolmogorov complexityrecursion theory
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 593 - 613
Copyright
Copyright © Association for Symbolic Logic 2008

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