Arithmetic Complexity, Kleene Closure, and Formal Power Series

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: There is a language in AC0 for which inversion and root extraction are GapL-complete, and there is a finite set for which inversion and root extraction are GapNC1-complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC1, such as GapAC0. We initiate work in this direction by classifying the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.