Representing Powerdomain Elements as Monadic Second Order Predicates

This report characterizes the powerdomain constructions which have been used in the semantics of programming languages in terms of formulas of first order logic under a preordering of provable implication. This provides an intuitive representation which suggests a new form of powerdomain - called the mixed powerdomain - which expresses data in a different way from the well-known constructions from programming semantics. It can be shown that the mixed powerdomain has many of the properties associated with the convex powerdomain such as the possibility of solving recursive equations and a simple algebraic characterization

Tensor Products and Powerspaces in Quantitative Domain Theory

AbstractOne approach to quantitative domain theory is the thesis that the underlying boolean logic of ordinary domain theory which assumes only values in the set {true, false} is replaced by a more elaborate logic with values in a suitable structure Ν. (We take Ν to be a value quantale.) So the order ⊑ is replaced by a generalised quasi-metric d, assigning to a pair of points the truth value of the assertion x ⊑ y.In this paper, we carry this thesis over to the construction of powerdomains. This means that we assume the membership relation ∈ to take its values in Ν. This is done by requiring that the value quantale Ν carries the additional structure of a semiring. Powerdomains are then constructed as free modules over this semiring.For the case that the underlying logic is the logic of ordinary domain theory our construction reduces to the familiar Hoare powerdomain. Taking the logic of quasi-metric spaces, i.e. Ν = [0, ∞] with usual addition and multiplication, reveals a close connection to the powerdomain of extended probability measures.As scalar multiplication need not be nonexpansive we develop the theory of moduli of continuity and m-continuous functions. This makes it also possible to consider functions between quantitative domains with different underlying logic. Formal union is an operation which takes pairs as input, so we investigate tensor products and their behavior with respect to the ideal completion

The Mixed Powerdomain

This paper introduces an operator M called the mixed powerdomain which generalizes the convex (Plotkin) powerdomain. The construction is based on the idea of representing partial information about a set of data items using a pair of sets, one representing partial information in the manner of the upper (Smyth) powerdomain and the other in the manner of the lower (Hoare) powerdomain where the components of such pairs are required to satisfy a consistency condition. This provides a richer family of meaningful partial descriptions than are available in the convex powerdomain and also makes it possible to include the empty set in a satisfactory way. The new construct is given a rigorous mathematical treatment like that which has been applied to the known powerdomains. It is proved that M is a continuous functor on bifinite domains which is left adjoint to the forgetful functor from a category of continuous structures called mix algebras. For a domain D with a coherent Scott topology, elements of M D can be represented as pairs (U, V) where U ⊆ D is a compact upper set, V ⊆ D is a closed set and the downward closure of U ∩ V is equal to V. A Stone dual characterization of M is also provided

Kantorovich-Rubinstein Quasi-Metrics IV: Lenses, Quasi-Lenses and Forks

International audienceLenses and quasi-lenses on a space X form models of erratic non-determinism. When X is equipped with a quasi-metric d, there are natural quasi-metrics d_P and d^a_P on the space of quasi-lenses on X, which resemble the Pompeiu-Hausdorff metric (and contain it as a subcase when d is a metric), and are tightly connected to the Kantorovich-Rubinstein quasi-metrics d_KR and d^a_KR of Parts I, II and III, through an isomorphism between quasi-lenses and socalled discrete normalized forks. We show that the space of quasi-lenses on X is continuous complete, resp. algebraic complete, if X, d is itself continuous complete, resp. algebraic complete. In those cases, we also show that the d_P-Scott and d^a_P-Scott topologies coincide with the Vietoris topology. We then prove similar results on spaces of (sub)normalized forks, not necessarily discrete; those are models of mixed erratic non-determinism and probabilistic choice. For that, we need the additional assumption that the cone LX of lower continuous maps from X to the extended non-negative real numbers, with the Scott topology, has an almost open addition map (which is the case if X is locally compact and coherent, notably); we also need X to be compact in the case of normalized forks. The relevant quasi-metrics are simple extensions of the Kantorovich-Rubinstein quasi-metrics d^a_KR