Subword complexity and decomposition of the set of factors

TL;DR

This paper introduces a hierarchy of language classes based on factor decomposition and explores their relationship with subword complexity, revealing that certain classes correspond to words with linear complexity and others have polynomial bounds.

Contribution

It defines the classes $W_k$ based on factor decomposition and establishes their connection with subword complexity, including characterizations for $W_2$ and strict inclusions for higher classes.

Findings

$W_2$ coincides with words of linear complexity.

$W_k$ is included in words of complexity $O(n^{k-1})$.

Inclusion is strict for $k > 2$.

Abstract

In this paper we explore a new hierarchy of classes of languages and infinite words and its connection with complexity classes. Namely, we say that a language belongs to the class $L_k$ if it is a subset of the catenation of $k$ languages $S_1\cdots S_k$, where the number of words of length $n$ in each of $S_i$ is bounded by a constant. The class of infinite words whose set of factors is in $L_k$ is denoted by $W_k$. In this paper we focus on the relations between the classes $W_k$ and the subword complexity of infinite words, which is as usual defined as the number of factors of the word of length $n$. In particular, we prove that the class $W_{2}$ coincides with the class of infinite words of linear complexity. On the other hand, although the class $W_{k}$ is included in the class of words of complexity $O(n^{k-1})$, this inclusion is strict for $k> 2$.