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The aperiodic Domino problem in higher dimension

Benjamin Hellouin de Menibus 1 Antonin Callard 1 
1 GALaC - Graphes, Algorithmes et Combinatoire
LISN - Laboratoire Interdisciplinaire des Sciences du Numérique, AAC - Algorithmes, Apprentissage et Calcul
Abstract : The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift. [8] proved that this problem is co-recursively enumerable (Π 1 0-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic (Σ 1 1-complete), in higher dimension: d ≥ 4 in the finite type case, d ≥ 3 for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity. This complexity jump is surprising for two reasons: first, it separates 2-and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.
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Submitted on : Monday, February 14, 2022 - 5:04:51 PM
Last modification on : Friday, August 5, 2022 - 9:27:32 AM

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Benjamin Hellouin de Menibus, Antonin Callard. The aperiodic Domino problem in higher dimension. 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), Mar 2022, Marseille, France. pp.1-15, ⟨10.4230/LIPIcs.STACS.2022.18⟩. ⟨hal-03573476⟩

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