The Complexity of Unavoidable Word Patterns

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.