The Aperiodic Domino Problem in Higher Dimension

Authors Antonin Callard , Benjamin Hellouin de Menibus

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Author Details

Antonin Callard
  • Université Paris-Saclay, ENS Paris-Saclay, Département Informatique, 91190 Gif-sur-Yvette, France
Benjamin Hellouin de Menibus
  • Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400 Orsay, France


The authors are grateful to the three referees for their many remarks and improvements. The first author is grateful to the Excellence group for being such cheerful co-interns and friends.

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Antonin Callard and Benjamin Hellouin de Menibus. The Aperiodic Domino Problem in Higher Dimension. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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. [Grandjean et al., 2018] proved that this problem is co-recursively enumerable (Π₀¹-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic (Σ₁¹-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Problems, reductions and completeness
  • Subshift
  • periodicity
  • aperiodicity
  • domino problem
  • subshift of finite type
  • sofic subshift
  • effective subshift
  • tilings
  • computability


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