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On Long Words Avoiding Zimin Patterns

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Abstract

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.

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Notes

  1. Our definition of strict prefix is slightly non-standard as ε is a strict prefix of any non-empty word.

  2. Again, remark that our definition is slightly non-standard as strict prefixes or strict suffixes are in general not strict infixes.

  3. http://vaucanson-project.org/Awali/

  4. Recall that there are two occurrences of p in [ [i] ]n+ 1.

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Authors and Affiliations

  1. Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEM, Marne-la-Vallée, France

    Arnaud Carayol

  2. Fachgebiet Theoretische Informatik / Komplexe Systeme Fachbereich Elektrotechnik / Informatik, Universität Kassel, Wilhelmshöher Allee 73, 34121, Kassel, Germany

    Stefan Göller

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  1. Arnaud Carayol
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  2. Stefan Göller
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Correspondence to Arnaud Carayol or Stefan Göller.

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Carayol, A., Göller, S. On Long Words Avoiding Zimin Patterns. Theory Comput Syst 63, 926–955 (2019). https://doi.org/10.1007/s00224-019-09914-2

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