Continuous L-domains in logical form

Abstract

We introduce a framework of approximable disjunctive propositional logic, which is the logic that results from a disjunctive propositional logic by adding an additional connective. The Lindenbaum algebra of this logic is an approximable dD-algebra. We show that for any approximable dD-algebra, its approximable filters ordered by set inclusion form a continuous L-domain. Conversely, every continuous L-domain can be represented as an approximable dD-algebra. Moreover, we establish a categorical equivalence between the category of approximable dD-algebras with approximable dD-algebra morphisms and that of continuous L-domains with Scott-continuous functions. This extends Abramsky's Domain Theory in Logical Form to the world of continuous L-domains. As an application, we give an affirmative answer to an open problem of Chen and Jung.