The Complexity of Unavoidable Word Patterns

TL;DR

This paper explores the complexity and prevalence of unavoidable word patterns over finite alphabets, providing bounds, algorithms, and complexity results for identifying such patterns.

Contribution

It introduces new bounds on avoiding words, quantifies the decline in unavoidable words with length, and offers algorithms and complexity analysis for their detection.

Findings

Unavoidable words become exponentially rarer as length increases.

Upper bounds are established for avoiding words of unavoidable patterns.

Algorithms and complexity results for detecting unavoidable words are provided.

Abstract

The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping the pattern into every word. We present various complexity-related properties of unavoidable words. For words that are unavoidable, we provide an upper bound to the lengths of words that avoid them. A natural subsequent question is how many unavoidable words there are. We show that the fraction of words that are unavoidable drops exponentially fast in the length of the word. This allows us to calculate an upper bound on the number of unavoidable patterns for any given finite alphabet. Subsequently, we investigate computational aspects of unavoidable words. In particular, we exhibit concrete algorithms for determining whether a word is unavoidable. We also prove results on the computational complexity of the problem of determining whether a given word is unavoidable.