Bounded complete domains and their logical form

Abstract

We establish a framework of approximable disjunctive sequent calculus, which is sound and complete with respect to approximable FD-algebras. We show that the category of approximable FD-algebras with approximable mappings is equivalent to that of bounded complete domains with Scott continuous functions. This extends Abramsky's logical representation of Scott domains as domain prelocales to a continuous setting. We also consider some domain constructions applied to approximable FD-algebras and show how to construct the approximable FD-algebras we need for defining the semantics of programming languages. According to a substructure relation, we define a pointed ω-chain complete class of approximable FD-algebras, on which the domain constructions are made continuous and then the initial solutions to recursive domain equations are transformed into the fixed-points of such continuous functions.