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On Domain Theory over Girard Quantales

Waszkiewicz P.

Fundamenta Informaticae

2009

Vol. 92, nr 1-2

169-192

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.

Domain Theory as a Tool for Topology - a Case Study

Waszkiewicz P.

Schedae Informaticae

2003

Vol. 12

17-26

In this paper we adopt a certain view on continuous posets and see them as models of their spaces of maximal elements, which are most often topologies rich in structure. Adopting this perspective seems to be fruitful: we are often able to match structural properties of the modelling poset to properties of the modelled space. It was discovered by Mike Reed and Keye Martin two years ago that existence of a measurement on the model corresponds to existence of a development for the modelled topological space. We present an elementary proof of this fact and show how one can use this result to give a new proof to one of the first metrization theorems in Topology.