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.