Resource-Bounded Kolmogorov Complexity Provides an Obstacle to Soficness of Multidimensional Shifts

Authors Julien Destombes, Andrei Romashchenko

Thumbnail PDF


  • Filesize: 0.52 MB
  • 17 pages

Document Identifiers

Author Details

Julien Destombes
  • LIRMM, University of Montpellier, Montpellier, France
Andrei Romashchenko
  • LIRMM, University of Montpellier, CNRS, Montpellier, France


We are indebted to Bruno Durand, Alexander Shen, and Ilkka Törmä for fruitful discussions. We are grateful to Pierre Guillon and Emmanuel Jeandel for motivating comments. We also thank the anonymous referees of STACS 2019 for many valuable comments.

Cite AsGet BibTex

Julien Destombes and Andrei Romashchenko. Resource-Bounded Kolmogorov Complexity Provides an Obstacle to Soficness of Multidimensional Shifts. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We suggest necessary conditions of soficness of multidimensional shifts formulated in terms of resource-bounded Kolmogorov complexity. Using this technique we provide examples of effective and non-sofic shifts on Z^2 with very low block complexity: the number of globally admissible patterns of size n x n grows only as a polynomial in n.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
  • Mathematics of computing → Combinatorics
  • Sofic shifts
  • Block complexity
  • Kolmogorov complexity


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


  1. N. Aubrun, S. Barbieri, and E. Jeandel. About the Domino Problem for Subshifts on Groups. Sequences, Groups, and Number Theory. Birkhäuser, Cham, pages 331-389, 2018. Google Scholar
  2. B. Durand, L. Levin, and A. Shen. Complex tilings. The Journal of Symbolic Logic, 73:2:593-613, 2008. Google Scholar
  3. F.C. Hennie and R.E. Stearns. Two-tape simulation of multitape Turing machines. Journal of the ACM, 13(4):533-546, 1966. Google Scholar
  4. E. Jeandel. Propriétés structurelles et calculatoires des pavages. Habilitation thesis, Université Montpellier 2, 2011. Google Scholar
  5. S. Kass and K. Madden. A sufficient condition for non-soficness of higher-dimensional subshifts. Proceedings of the American Mathematical Society, 141:11:3803-3816, 2013. Google Scholar
  6. A. N. Kolmogorov. Three approaches to the quantitative definition of information. Problems of information transmission, 1:1:1-7, 1965. Google Scholar
  7. M. Li and P. Vitányi. An introduction to Kolmogorov complexity and its applications. 3rd ed. Springer, New York, 2008. Google Scholar
  8. D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1995. Google Scholar
  9. M.Morse and G.A. Hedlund. Symbolic dynamics. Amer. J. Math., 60:815-866, 1938. Google Scholar
  10. M. Morse and G. A. Hedlund. Symbolic dynamics II: Sturmian trajectories. Amer. J. Math., 62:1-42, 1940. Google Scholar
  11. S. Mozes. Tilings, substitution systems and dynamical systems generated by them. Journal d'analyse mathématique (Jerusalem), 53:139-186, 1989. Google Scholar
  12. N. Ormes and R. Pavlov. Extender sets and multidimensional subshifts. Ergodic Theory and Dynamical Systems, 36:3:908-923, 2016. Google Scholar
  13. R. Pavlov. A class of nonsofic multidimensional shift spaces. Proceedings of the American Mathematical Society, 141:3:987-996, 2013. Google Scholar
  14. A. Rumyantsev and M. Ushakov. Forbidden substrings, Kolmogorov complexity and almost periodic sequences. In Proc. Annual Symposium on Theoretical Aspects of Computer Science, pages 396-407, 2006. Google Scholar
  15. V. Salo. Subshifts with sparse projective subdynamics. arXiv preprint, arXiv:1605.09623, 2016. Google Scholar
  16. B. Weiss. Subshifts of finite type and sofic systems. Monatsh. Math., 77:462-474, 1973. Google Scholar
  17. L.B. Westrick. Seas of squares with sizes from a Π⁰₁ set. Israel Journal of Mathematics, 22:1, 2017. Google Scholar
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