53 research outputs found
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
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
Relation lifting, with an application to the many-valued cover modality
We introduce basic notions and results about relation liftings on categories
enriched in a commutative quantale. We derive two necessary and sufficient
conditions for a 2-functor T to admit a functorial relation lifting: one is the
existence of a distributive law of T over the "powerset monad" on categories,
one is the preservation by T of "exactness" of certain squares. Both
characterisations are generalisations of the "classical" results known for set
functors: the first characterisation generalises the existence of a
distributive law over the genuine powerset monad, the second generalises
preservation of weak pullbacks. The results presented in this paper enable us
to compute predicate liftings of endofunctors of, for example, generalised
(ultra)metric spaces. We illustrate this by studying the coalgebraic cover
modality in this setting.Comment: 48 pages, accepted for publication in LMC
Suggestions for a non-monotonic feature logic
We use Scott's domain theory and methods from Reiter's default logic to suggest some ways of modelling default constraints in feature logic. We show how default feature rules, derived from default constraints, can be used to give ways to augment strict feature structures with default information
A domain equation for refinement of partial systems
Published versio
A representable approach to finite nondeterminism
AbstractWe reformulate denotational semantics for nondeterminism, taking a nondeterministic operation V on programs, and sequential composition, as primitive. This gives rise to binary trees. We analyse semantics for both type and program constructors such as products and exponential types, conditionals and recursion, in this setting. In doing so, we define new category-theoretic structures, in particular premonoidal categories. We also account for equivalences of programs such as those induced by associativity, symmetry and idempotence of V, and we study finite approximation by enrichment over the category of ω-cpos with least element. We also show how to recover the classical powerdomains, especially the convex powerdomain, as three instances of a general, computationally natural, construction
- …