The Mixed Powerdomain

This paper 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. The goal is to reveal the basic logical significance of the powerdomain elements by casting them in the right setting. Such a treatment may contribute to a better understanding of their potential uses in areas which deal with concepts of sets and partial information such as databases and computational linguistics. This way of viewing powerdomain elements 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 is shown that the mixed powerdomain has many of the properties associated with the convex powerdomain such as the possiblity of solving recursive equations and a simple algebraic characterization

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

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

A Universal Characterization of the Double Powerlocale

This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321. The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc

Free dcpo-algebras via directed spaces

Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. We will show that the D-completion of free algebras over a Scott space $\Sigma L$, on the context of directed spaces, are exactly the free dcpo-algebras over dcpo $L$, which reveals the close connection between directed powerspaces and powerdomains. By this result, we provide a topological representation of upper, lower and convex powerdomains of dcpos uniformly.Comment: 18 page

On F

We introduce a new construction—FS+-domain—and prove that the category with FS+-domains as objects and Scott continuous functions as morphisms is a Cartesian closed category. We obtain that the Plotkin powerdomain PP(L) over an FS-domain L is an FS+-domain