On Domain Theory over Girard Quantales

This paper is about a generalization of Scott's domain theory in such a way that its definitions and theorems become meaningful in quasimetric spaces. The generalization is achieved by a change of logic: the fundamental concepts of original domain theory (order, way-below relation, Scott-open sets, continuous maps, etc.) are interpreted as predicates that are valued in an arbitrary completely distributive Girard quantale (a CDG quantale). Girard quantales are known to provide a sound and complete semantics for commutative linear logic, and complete distributivity adds a notion of approximation to our setup. Consequently, in this paper we speak about domain theory based on commutative linear logic with some additional reasoning principles following from approximation between truth values. Concretely, we: (1) show how to define continuous Q-domains, i.e. continuous domains over a CDG quantale Q; (2) study their way-below relation, and (3) study the rounded ideal completion of Q-abstract bases. As a case study, we (4) demonstrate that the domain-theoretic construction of the Hoare, Smyth and Plotkin powerdomains of a continuous dcpo can be straightforwardly adapted to yield corresponding constructions for continuous Q-domains.