Arithmetic Complexity, Kleene Closure, and Formal Power Series

Abstract

In this paper we show how two fundamental operations used in formal language theory provide useful tools for the investigation of arithmetic complexity classes. More precisely, we use Kleene closure of languages and inversion of formal power series to investigate subclasses of the complexity class GapL. GapL is the complexity class that characterizes the complexity of computing the determinant; it corresponds to counting the number of accepting and rejecting paths of nondeterministic logspace-bounded Turing machines.) We define a counting version of Kleene closure and show that it is intimately related to inversion within the complexity classes GapL and GapNC^1. In particular, we prove that Kleene closure and inversion are both hard operations in the following sense There is a set in AC^0 for which Kleene closure is NL-complete and inversion is GapL-complete. There is a finite set for which Kleene closure is NC^1-complete and inversion is GapNC^1-complete. Furthermore, we classify 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.