Abstract
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove the existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the fixed point construction [12]; we improve this general technique and make it enforce the property of local regularity of tilings needed for quasiperiodicity. We prove also a stronger result: any \(\mathrm \Pi _1^0\)-class can be recursively transformed into a tile set so that the Turing degrees of the resulting tilings consists exactly of the upper cone based on the Turing degrees of the latter.
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Acknowledgements
We thank Laurent Bienvenu and Emmanuel Jeandel for many prolific discussions. We are also very grateful to the three anonymous referees for exceptionally detailed and instructive comments.
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Durand, B., Romashchenko, A. (2015). Quasiperiodicity and Non-computability in Tilings. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_17
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DOI: https://doi.org/10.1007/978-3-662-48057-1_17
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