Theory of Computing Systems

, Volume 63, Issue 5, pp 926–955 | Cite as

On Long Words Avoiding Zimin Patterns

  • Arnaud CarayolEmail author
  • Stefan GöllerEmail author
Part of the following topical collections:
  1. Special Issue on Theoretical Aspects of Computer Science (STACS 2017)


A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n − 3 exponentials, even for k = 2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.


Unavoidable patterns Combinatorics on words Lower bounds 



  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley (2015)Google Scholar
  2. 2.
    Bėal, M.-P., Carton, O., Prieur, C., Sakarovitch, J.: Squaring transducers: an efficient procedure for deciding functionality and sequentiality. Theor. Comput. Sci. 292(1), 45–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carayol, A., Göller, S.: On long words avoiding zimin patterns. In: Vollmer, H., Vallėe, B. (eds.) 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, volume 66 of LIPIcs, pp. 19:1–19:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  4. 4.
    Carayol, A., Gȯller, S.: On long words avoiding Zimin patterns. CoRR, arXiv:1902.05540 (2019)
  5. 5.
    Conlon, D., Fox, J., Sudakov, B.: Tower-type bounds for unavoidable patterns in words (2017)Google Scholar
  6. 6.
    Cooper, J., Rorabaugh, D.: Bounds on Zimin word avoidance. CoRR, arXiv:1409.3080 (2014)
  7. 7.
    Cooper, J., Rorabaugh, D.: Asymptotic density of Zimin words. Discret. Math. Theor. Comput. Sci. 18, 3 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Currie, J.D.: Pattern avoidance: themes and variations. Theor. Comput. Sci. 339(1), 7–18 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Currie, J.D., Linek, V.: Avoiding patterns in the Abelian sense. Canadian J. Math. 51(4), 696–714 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bean, G.F., McNulty, D.R., Ehrenfeucht, A.: Avoidable patterns in strings of symbols. Pac. J. Math. 85, 261–294 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Jancar, P.: Equivalences of pushdown systems are hard. In: Proceedings of FOSSACS 2014, Volume 8412 of Lecture Notes in Computer Science, pp. 1–28. Springer (2014)Google Scholar
  13. 13.
    Jugé, V: Abelian ramsey length and asymptotic lower bounds. CoRR, arXiv:1609.06057 (2016)
  14. 14.
    Reinhardt, K.: The complexity of translating logic to finite automata. In: Automata, Logics, and Infinite Games: A Guide to Current Research, Volume 2500 of Lecture Notes in Computer Science, pp. 231–238. Springer (2001)Google Scholar
  15. 15.
    Rorabaugh, D.: Toward the combinatorial limit theory of free words. CoRR, arXiv:1509.04372 (2015)
  16. 16.
    Rytter, W., Shur, A.M.: Searching for Zimin patterns. Theor Comput. Sci. 571, 50–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sapir, M.V.: Combinatorics on words with applications. Technical report (1995)Google Scholar
  18. 18.
    Schmitz, S.: Complexity hierarchies beyond elementary. CoRR, arXiv:1312.5686 (2014)
  19. 19.
    Sėnizergues, G.: The equivalence problem for t-turn DPDA is co-np. In: Proceedings of ICALP, Volume 2719 of Lecture Notes in Computer Science, pp. 478–489. Springer (2003)Google Scholar
  20. 20.
    Stirling, C.: Deciding DPDA equivalence is primitive recursive. In: Proceedings of ICALP 2002, Volume 2380 of Lecture Notes in Computer Science, pp. 821–832. Springer (2002)Google Scholar
  21. 21.
    Stockmeyer, L.J.: The Complexity of Decision Problems in Automata and Logic. PhD thesis, Massachusetts Institute of Technology, Cambridge MA (1974)Google Scholar
  22. 22.
    Tao, J.: Pattern occurrence statistics and applications to the ramsey theory of unavoidable patterns. CoRR, arXiv:1406.0450 (2014)
  23. 23.
    Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Skrifter I Mat.-Nat. Kl. Christiania 7, 1–22 (1906)zbMATHGoogle Scholar
  24. 24.
    Zimin, A.I.: Blocking sets of terms. Math. USSR Sbornik 47, 50–57 (1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEMMarne-la-ValléeFrance
  2. 2.Fachgebiet Theoretische Informatik / Komplexe Systeme Fachbereich Elektrotechnik / InformatikUniversität KasselKasselGermany

Personalised recommendations