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The Complexity of Unavoidable Word Patterns

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Abstract

The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. The word α over a finite set A is said to be unavoidable for an infinite set B+ of nonempty words over a finite set B if, for all but finitely many elements w of B+, there exists a semigroup morphism \(\phi :A^{+}\rightarrow B^{+}\) such that ϕ(α) is a factor of w. We discuss unavoidability in the milieu of various types of complexity. For words that are unavoidable, we provide a constructive upper bound to the lengths of words that can avoid them. We then discuss the relative density of unavoidable words. Subsequently, we investigate computational aspects of unavoidable words, focusing on the computational complexity of determining whether a word is unavoidable. This culminates in a proof that this problem is NP-complete.

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Acknowledgements

My deepest thanks go to Willem Fouché, Petrus Potgieter, Pascal Vanier, Arno Pauly, Holger Spakowski, Niko Sauer, Narad Rampersad and James Currie. Without their generous patience, deep insight, diligence and kindness, this paper would not have been possible. Many thanks also to the wonderful anonymous editors and reviewers of TOCS. This paper has been written in partial fulfillment of the requirements for the degree Doctor of Philosophy in Operations Research at the University of South Africa.

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  1. University of South Africa, Department of Decision Sciences, Unisa Muckleneuk Campus Preller Street Muckleneuk, Pretoria, South Africa

    Paul Sauer

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Correspondence to Paul Sauer.

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Sauer, P. The Complexity of Unavoidable Word Patterns. Theory Comput Syst 66, 802–820 (2022). https://doi.org/10.1007/s00224-022-10090-z

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