License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.19
URN: urn:nbn:de:0030-drops-158296
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15829/
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Callard, Antonin ; Hellouin de Menibus, Benjamin

The Aperiodic Domino Problem in Higher Dimension

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LIPIcs-STACS-2022-19.pdf (0.9 MB)

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.
[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.

BibTeX - Entry

@InProceedings{callard_et_al:LIPIcs.STACS.2022.19,
  author =	{Callard, Antonin and Hellouin de Menibus, Benjamin},
  title =	{{The Aperiodic Domino Problem in Higher Dimension}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{19:1--19:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15829},
  URN =		{urn:nbn:de:0030-drops-158296},
  doi =		{10.4230/LIPIcs.STACS.2022.19},
  annote =	{Keywords: Subshift, periodicity, aperiodicity, domino problem, subshift of finite type, sofic subshift, effective subshift, tilings, computability}
}

Keywords: Subshift, periodicity, aperiodicity, domino problem, subshift of finite type, sofic subshift, effective subshift, tilings, computability
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022

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