Subword Complexity and Decomposition of the Set of Factors

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 Lk if it is a subset of the catenation of k languages S1 · · ·Sk, where the number of words of length n in each of Si is bounded by a constant. The class of infinite words whose set of factors is in Lk is denoted by Wk. In this paper we focus on the relations between the classesWk 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 W2 coincides with the class of infinite words of linear complexity. On the other hand, although the class Wk is included in the class of words of complexity O(nk−1), this inclusion is strict for k > 2.