Difference hierarchies and duality with an application to formal languages

Abstract

The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. Lattice theoretically, the difference hierarchy over a distributive lattice stratifies the Boolean algebra generated by it according to the minimum length of difference chains required to describe the Boolean elements. While each Boolean element is given by a finite difference chain, there is no canonical such writing in general. We show that, relative to the filter completion, or equivalently, the lattice of closed upsets of the dual Priestley space, each Boolean element over the lattice has a canonical minimum length decomposition into a Hausdorff difference chain. As a corollary, each Boolean element over a co-Heyting algebra has a canonical difference chain (and an order dual result holds for Heyting algebras). With a further generalization of this result involving a directed family of closure operators on a Boolean algebra, we give an elementary proof of the fact that if a regular language is given by a Boolean combination of universal sentences using arbitrary numerical predicates then it is also given by a Boolean combination of universal sentences using only regular numerical predicates.