Low-Complexity Tilings of the Plane

  • Jarkko KariEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)


A two-dimensional configuration is a coloring of the infinite grid \(\mathbb {Z}^2\) with finitely many colors. For a finite subset D of \(\mathbb {Z}^2\), the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.


Pattern complexity Periodicity Nivat’s conjecture Low complexity configurations Low complexity subshifts Commutative algebra Algebraic subshifts Domino problem 


  1. 1.
    Axenovich, M.A.: On multiple coverings of the infinite rectangular grid with balls of constant radius. Discrete Math. 268(1), 31–48 (2003). Scholar
  2. 2.
    Bhattacharya, S.: Periodicity and decidability of tilings of \(\mathbb{Z}^{2}\). CoRR abs/1602. 05738 (2016).
  3. 3.
    Birkhoff, G.D.: Quelques théorèmes sur le mouvement des systèmes dynamiques. Bull. Soc. Math. France 40, 305–323 (1912). Scholar
  4. 4.
    Cassaigne, J.: Subword complexity and periodicity in two or more dimensions. In: Rozenberg, G., Thomas, W. (eds.) Developments in Language Theory. Foundations, Applications, and Perspectives, pp. 14–21. World Scientific (1999)Google Scholar
  5. 5.
    Cyr, V., Kra, B.: Nonexpansive \({\mathbb{Z}}^2\)-subdynamics and Nivat’s Conjecture. Trans. Amer. Math. Soc. 367(9), 6487–6537 (2015). Scholar
  6. 6.
    Kari, J., Moutot, E.: Decidability and periodicity of low complexity tilings. CoRR abs/1904.01267 (2019).
  7. 7.
    Kari, J., Moutot, E.: Nivat’s conjecture and pattern complexity in algebraic subshifts. Theoret. Comput. Sci. (2019, to appear).
  8. 8.
    Kari, J., Szabados, M.: An algebraic geometric approach to multidimensional words. In: Maletti, A. (ed.) CAI 2015. LNCS, vol. 9270, pp. 29–42. Springer, Cham (2015). Scholar
  9. 9.
    Kari, J., Szabados, M.: An algebraic geometric approach to Nivat’s conjecture. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 273–285. Springer, Heidelberg (2015). Scholar
  10. 10.
    Kari, J., Szabados, M.: An algebraic geometric approach to Nivat’s conjecture. CoRR abs/1605.05929 (2016).
  11. 11.
    Lagarias, J.C., Wang, Y.: Tiling the line with translates of one tile. Invent. Math. 124, 341–365 (1996). Scholar
  12. 12.
    Nivat, M.: Keynote address at the 25th anniversary of EATCS, during ICALP (1997)Google Scholar
  13. 13.
    Schmidt, K.: Dynamical systems of algebraic origin. Progress in mathematics. Birkhäuser, Basel (1995).
  14. 14.
    Szabados, M.: Nivat’s conjecture holds for sums of two periodic configurations. In: Tjoa, A.M., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds.) SOFSEM 2018. LNCS, vol. 10706, pp. 539–551. Springer, Cham (2018). Scholar
  15. 15.
    Szegedy, M.: Algorithms to tile the infinite grid with finite clusters. In: Proceedings 39th Annual Symposium on Foundations of Computer Science, FOCS 1998, pp. 137–147. IEEE Computer Society (1998).
  16. 16.
    Wang, H.: Proving theorems by pattern recognition - II. Bell Syst. Tech. J. 40(1), 1–41 (1961). Scholar

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations