Document Open Access Logo

Decidability and Periodicity of Low Complexity Tilings

Authors Jarkko Kari , Etienne Moutot

Thumbnail PDF


  • Filesize: 0.58 MB
  • 12 pages

Document Identifiers

Author Details

Jarkko Kari
  • University of Turku, Finland
Etienne Moutot
  • University of Turku, Finland
  • Université de Lyon - ENS de Lyon - UCBL - CNRS - LIP, France

Cite AsGet BibTex

Jarkko Kari and Etienne Moutot. Decidability and Periodicity of Low Complexity Tilings. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 14:1-14:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


In this paper we study low-complexity colorings (or tilings) of the two-dimensional grid ℤ². A coloring is said to be of low complexity with respect to a rectangle if there exists m,n∈ℕ such that there are no more than mn different rectangular m× n patterns in it. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. Suppose we are given at most nm rectangular patterns of size n× m. If Nivat’s conjecture is true, one can only build periodic colorings out of these patterns - meaning that if the m× n rectangular patterns of the coloring are among these mn patterns, it must be periodic. The main contribution of this paper proves that there exists at least one periodic coloring build from these patterns. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Finally, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. The results also extend to other convex shapes in place of the rectangle.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Computability
  • Nivat’s conjecture
  • domino problem
  • decidability
  • low pattern complexity
  • 2D subshifts
  • symbolic dynamics


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. R. Berger. The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society. American Mathematical Society, 1966. Google Scholar
  2. S. Bhattacharya. Periodicity and decidability of tilings of ℤ². ArXiv e-prints, February 2016. URL:
  3. George D. Birkhoff. Quelques théorèmes sur le mouvement des systèmes dynamiques. Bulletin de la Société Mathématique de France, 40:305-323, 1912. URL:
  4. Mike Boyle and Douglas Lind. Expansive subdynamics. Transactions of the American Mathematical Society, 349(1):55-102, 1997. URL:
  5. Julien Cassaigne. Subword complexity and periodicity in two or more dimensions. In Grzegorz Rozenberg and Wolfgang Thomas, editors, Developments in Language Theory. Foundations, Applications, and Perspectives. Aachen, Germany, 6-9 July 1999, pages 14-21. World Scientific, 1999. URL:
  6. T. Ceccherini-Silberstein and M. Coornaert. Cellular Automata and Groups. Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2010. Google Scholar
  7. Van Cyr and Bryna Kra. Nonexpansive ℤ²-subdynamics and Nivat’s Conjecture. Transactions of the American Mathematical Society, 367(9):6487-6537, February 2015. URL:
  8. Jarkko Kari and Etienne Moutot. Nivat’s conjecture and pattern complexity in algebraic subshifts. Theoretical Computer Science, 2019. URL:
  9. Jarkko Kari and Michal Szabados. An Algebraic Geometric Approach to Nivat’s Conjecture. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, pages 273-285. Springer, 2015. Google Scholar
  10. M. Nivat. Keynote address at the 25th anniversary of EATCS, during ICALP 1997, Bologna, 1997. Google Scholar
  11. Michal Szabados. Nivat’s conjecture holds for sums of two periodic configurations. In A Min Tjoa, Ladjel Bellatreche, Stefan Biffl, Jan van Leeuwen, and Jiří Wiedermann, editors, SOFSEM 2018: Theory and Practice of Computer Science, pages 539-551, Cham, 2018. Springer International Publishing. Google Scholar
  12. Mario Szegedy. Algorithms to tile the infinite grid with finite clusters. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 137-147. IEEE Computer Society, 1998. URL:
  13. H. Wang. Proving theorems by pattern recognition - II. The Bell System Technical Journal, 40(1):1-41, 1961. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail