Abstract not found

Abstract not found

Abstract not found

We study fixpoints of operators on lattices. To this end

Abstract not found

Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98], which finds its roots in work by Lawvere ([Law73], cf. also [Wag94]) and which is related to early work by Stoltenberg (e.g. [Sto67], [Sto67a] and [FG84]), and the Smyth completion ([Smy89],[Smy91],[Smy94],[Sun93] and [Sun95]). A net-version of the Yoneda completion, complementing the net-version of the Smyth completion ([Sun95]), is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98]: "An interesting question is to characterize the family of generalized metric spaces for which [the Yoneda] completion is idempotent (it contains at least all ordinary metric spaces)." We show that the largest class of quasi-metric spaces idempotent under the Yoneda completion is precisely the class of Smyth-completable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]

. We study relations between predicate transformers and multifunctions in a topological setting based on closure operators. We give topological definitions of safety and liveness predicates and using these predicates we define predicate transformers. State transformers are multifunctions with values in the collection of fixed points of a closure operator. We derive several isomorphisms between predicate transformers and multifunctions. By choosing different closure operators we obtain multifunctions based on the usual power set construction, on the Hoare, Smyth and Plotkin power domains, and based on the compact and closed metric power constructions. Moreover, they are all related by isomorphisms to the predicate transformers. 1 Introduction There are (at least) two different ways of assigning a denotational semantics to a programming language: forward or backward. A typical forward semantics is a semantics that models a program as a function from initial states to final states. In th...

We show that every locally finite continuous valuation defined on the lattice of open sets of a regular or locally compact sober space extends uniquely to a Borel measure.

Hoare and He’s Unifying Theories of Programming take a relational view on semantics. The meaning of a non-deterministic, imperative program is described by ‘designs’ composed of two relations. They represent terminating states and relate the initial and final values of the observable variables, respectively. Several ‘healthiness conditions’ are imposed by the theory to obtain properties found in practice. This work determines the structure of designs and modifies the theory to support nonstrict computations. It achieves these goals by identifying healthiness conditions and related axioms that involve unnecessary restrictions and subsequently removing them. The outcome provides a clear account of the algebraic foundations of the Unifying Theories of Programming. One of the results is a generalisation of designs by constructing them on semirings with ideals, structures having fewer axioms than relations. This clarifies the essential algebraic structure of designs, allows the reuse of existing mathematical theory and connects to further semantical approaches. The framework is extended by algebraic formulations of finite and infinite iteration, domain, pre-image, determinacy, invariants and convergence. Calculations

Let F be a monotone operator on the complete lattice L into itself. Tarski's lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F is the image of L by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for F. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semicontinuity hypotheses. 1 * Introduction. Let L(£, J_, T, (J, Π) be a nonempty complete lattice with partial ordering Q, least upper bound U, greatest