Recurrent Partial Words

TL;DR

This paper explores the properties of infinite partial words with wildcards, focusing on their subword complexity and recurrence, revealing new possibilities not present in full words.

Contribution

It introduces the concept of recurrence for infinite partial words and links it to subword complexity, expanding understanding beyond traditional full words.

Findings

Partial words can have subword complexities unattainable by full words

Recurrence in partial words is characterized and connected to subword complexity

New classes of infinite words with unique properties are identified

Abstract

Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite partial word w, the number of distinct full words over the alphabet that are compatible with factors of w of length n, called subwords of w, refers to a measure of complexity of infinite partial words so-called subword complexity. This measure is of particular interest because we can construct partial words with subword complexities not achievable by full words. In this paper, we consider the notion of recurrence over infinite partial words, that is, we study whether all of the finite subwords of a given infinite partial word appear infinitely often, and we establish connections between subword complexity and recurrence in this more general framework.