Achim Jung

Using powerdomains to generalize relational databases

Abstract Much of relational algebra and the underlying principles of relational database design have a simple representation in the theory of domains that is traditionally used in the denotational semantics of programming languages. By investigating the possible orderings on powerdomains that are well known in the study of nondeterminism and concurrency it is possible to show that many of the ideas in relational databases apply to structures that are much more general than relations. This also suggests a method of representing database objects as typed objects in programming languages. In this paper we show how operations such as natural join and projection —which are fundamental to relational database design—can be generalized, and we use this generalized framework to give characterizations of several relational database concepts including functional dependencies and universal relations. All of these have a simple-minded semantics in terms of the underlying domains, which can be thought of as domains of partial descriptions of “real-world” objects. We also discuss the applicability of relational database theory to nonrelational structures such as records with variants, higher-order relations, recursive structures and other ordered spaces.

The classification of continuous domains

The long-standing problem of finding the maximal Cartesian closed categories of continuous domains is solved. The solution requires the definition of a new class of continuous domains, called FS-domains, which contains all retracts of SFP-objects. The properties of FS-domains are discussed.<<ETX>>

A New Characterization of Lambda Definability

We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for hereditarily finite Henkin models remains an open problem. But if the variable set allowed in terms is also restricted to be finite then our techniques lead to a decision procedure.

Stably Compact Spaces and the Probabilistic Powerspace construction

We put forward a revised deflnition of stably compact spaces which allows us to show their equivalence with Nachbin’s compact ordered spaces in an entirely elementary fashion. We then exhibit some constructions for stably compact spaces which apparently have not appeared in the literature before. These constructions allow us to show that the set of (sub-)probability valuations can be equipped with a topology which turns this set into another stably compact space. The topology chosen is not random; it is the weakest topology which makes integration of lower semicontinuous functions a continuous operation.

On the Duality of Compact vs. Open

It is a pleasant fact that Stone‐duality may be described very smoothly when restricted to the category of compact spectral spaces: The Stone‐duals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations.

The probabilistic powerdomain for stably compact spaces

Abstract This paper reviews the one-to-one correspondence between stably compact spaces (a topological concept covering most classes of semantic domains) and compact ordered Hausdorff spaces. The correspondence is extended to certain classes of real-valued functions on these spaces. This is the basis for transferring methods and results from functional analysis to the non-Hausdorff setting. As an application of this, the Riesz Representation Theorem is used for a straightforward proof of the (known) fact that every valuation on a stably compact space extends uniquely to a Radon measure on the Borel algebra of the corresponding compact Hausdorff space. The view of valuations and measures as certain linear functionals on function spaces suggests considering a weak topology for the space of all valuations. If these are restricted to the probabilistic or sub-probabilistic case, then another stably compact space is obtained. The corresponding compact ordered space can be viewed as the set of (probability or sub-probability) measures together with their natural weak topology.

Studying the Fully Abstract Model of PCF within its Continuous Function Model

We give a concrete presentation of the inequationally fully abstract model of PCF as a continuous projection of the inductively reachable subalgebra of PCFs continuous function model.

Cartesian closed categories of algebraic CPOs

Abstract The results presented in this paper were inspired by the work of Smyth, who showed that there is a largest cartesian closed full subcategory inside the category of countably based algebraic cpos with least element, namely the category of profinite domains. Removing the countability condition, we show that there are exactly two maximal cartesian closed full subcategories inside the category of algebraic cpos with least element. One is the natural extension of the class of profinite domains to the uncountable case, the other is a new class of domains, which are characterized by the property that every principal ideal is a complete lattice. The name L- domain is introduced for cpos with this property. Passing to the general situation where no least element is required anymore, we find a pair of categories in place of each the profinite domains and the algebraic L-domains. Thus there are four maximal cartesian closed categories of algebraic cpos in this case. This complete overview over the possible classes of domains allows to prove general theorems about them. This is illustrated by the result that a cpo has an algebraic function space if and only if its space of strict continuous functions is algebraic.

Multi lingual sequent calculus and coherent spaces

We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalised in this fashion. The usual cut rule does not make sense for sequents which connect different logical systems because it mixes formulas from antecedent and succedent. We propose a different cut rule which addresses this problem. The new cut rule can be used as a basis for composition in a suitable category of logical systems. As it turns out, this category is equivalent to coherent spaces with certain relations between them. Finally, cut elimination in this set-up can be employed to provide a new explanation of the domain constructions in Samson Abramskys Domain Theory in Logical Form.

Stably Compact Spaces and Closed Relations

Abstract Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain “closed relations” and study the resulting categorical properties. Apart from extending ordinary continuous maps, these morphisms have a number of pleasing properties, the most prominent, perhaps, being that they correspond to preframe homomorphisms on the localic side. We exploit this Stone-type duality to establish that the category of stably compact spaces and closed relations has bilimits.