Summary: Arithmetic Complexity, Kleene Closure, and Formal
April 2, 2003
The aim of this paper is to use formal power series techniques to study the structure
of small arithmetic complexity classes such as GapNC1 and GapL. More precisely, we
apply 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 GapNC1 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 Kleene closure is NLOG-complete, and there is a finite set
for which inversion and root extraction are GapNC1-complete and Kleene closure is
NC1-complete, with respect to appropriate reducibilities.
The latter result raises the question of classifying finite languages so that their in-
verses 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.