Subword Complexity and Decomposition of the Set of Factors

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 $\mathcal L_k$ if it is a subset of the catenation of k languages S ₁ ⋯ 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 $\mathcal L_k$ is denoted by $\mathcal W_k$. In this paper we focus on the relations between the classes $\mathcal W_k$ 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 $\mathcal W_{2}$ coincides with the class of infinite words of linear complexity. On the other hand, although the class $\mathcal W_{k}$ is included in the class of words of complexity O(n ^k − 1), this inclusion is strict for k > 2.