On Long Words Avoiding Zimin Patterns

Arnaud Carayol (1) , Stefan Göller (2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LSV - Laboratoire Spécification et Vérification [Cachan]

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. A pattern is a finite word over some set of pattern variables. A pattern matches a word if the word can be obtained by substituting each variable appearing in the pattern by a non-empty word. The pattern xx matches the word nana when x is replaced by the word na. A word encounters a pattern if the pattern matches some infix of the word. For example, the word banana encounters the pattern xx (as the word nana is one of its infixes). The pattern xyx is encountered in precisely those words that contain two non-consecutive occurrences of the same letter, as e.g., the word abca. A pattern is unavoidable if over every finite alphabet every sufficiently long word encounters the pattern. Equivalently, by Kőnig's Lemma, a pattern is unavoidable if over every finite alphabet all infinite words encounter the pattern. If it is not the case, the pattern is said to be avoidable. The pattern xyx is easily seen to be unavoidable since every sufficiently long word over a non-empty finite alphabet must contain two non-consecutive occurrences of the same letter. On the other hand, the pattern xx is avoidable as Thue [19] gave an infinite word over a ternary alphabet that does not encounter the pattern xx. A precise characterization of unavoidable patterns was found by Zimin [20] and independently by Bean, Ehrenfeucht and McNulty [6], see also [13] for a more recent proof. This * Supported by Labex Digicosme, Univ. Paris-Saclay, project VERICONISS.

Domaines: Théorie et langage formel [cs.FL]

Arnaud Carayol, Stefan Göller. On Long Words Avoiding Zimin Patterns. 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), Mar 2017, Hannover, Germany. pp.19:1-19:13, ⟨10.4230/LIPIcs.STACS.2017.19⟩. ⟨hal-01803632⟩