Computing (or not) Quasi-Periodicity Functions of Tilings

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Computing (or not) Quasi-Periodicity Functions of Tilings
Abstract. We know that tilesets that can tile the plane always admit a quasiperiodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasiperiodic tiling. We prove that the tilings by a tileset that admits only quasiperiodic tilings have a recursively (and uniformly) bounded quasi-periodicity function. This corrects an error from [6, theorem 9] which stated the contrary. Instead we construct a tileset for which any quasi-periodic tiling has a quasi-periodicity function that cannot be recursively bounded. We provide such a construction for 1-dimensional effective subshifts and obtain as a corollary the result for tilings of the plane via recent links between these objects [1, 10]. Tilings of the discrete plane as studied nowadays have been introduced by Wang in order to study the decidability of a subclass of first order logic [26, 27, 5]. After Berger proved the undecidability of the dom...
Alexis Ballier, Emmanuel Jeandel
Added 01 Mar 2011
Updated 01 Mar 2011
Type Journal
Year 2010
Where CORR
Authors Alexis Ballier, Emmanuel Jeandel
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