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. From a lattice theoretic point of view, the difference hierarchy over a bounded 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. As a corollary each Boolean element over a (co-)Heyting algebra has a canonical difference chain. With a further generalization of this result involving a directed family of adjunctions with meetsemilattices, we give an elementary proof of the fact that a regular language is given by a Boolean combination of purely universal sentences using arbitrary numerical predicates if and only if it is given by a Boolean combination of purely universal sentences using only regular numerical predicates.

Stone duality for spectral sheaves and the patch monad

“…In the proof of Lemma 4.4, the initial input is the number of elements of L needed in the Boolean expressions for the individual b i . See [8] for an inductive construction of "difference chains" of minimal length for the elements of L B .…”

Probability over Plonka sums of Boolean algebras: states, metrics and topology

“…It is useful to recall that the present use of the term "Booleanisation" differs from other usages in literature, in lattice theory (see[13]) and in the theory of Boolean (inverse) semigroups (see[27]). …”

“…A topological space (X, T ) is Alexandrov discrete when the arbitrary intersection of open sets is an open set 13. A similar idea is developed from the construction of horizontal sums in[2].…”