The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:

We develop a compositional proof-system for the partial correctness of concurrent constraint programs. Soundness and (relative) completeness of the system are proved with respect to a denotational semantics based on the notion of strongest postcondition. The strongest postcondition semantics provides a justification of the declarative nature of concurrent constraint programs, since it allows to view programs as theories in the specification logic. 1 Introduction Concurrent constraint programming ([24, 25, 26]) (ccp, for short) is a concurrent programming paradigm which derives from replacing the store-as-valuation conception of von Neumann computing by the storeas -constraint model. Its computational model is based on a global store, represented by a constraint, which expresses some partial information on the values of the variables involved in the computation. The concurrent execution of different processes, which interact through the common store, refines the partial information of...

. We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. The category of cpos is contained as a full subcategory that is preserved by lifting, sums, products and function spaces. The construction of the powerdomain uses a cpo of binary trees, these being intensional representations of nondeterministic computation. The powerdomain is characterized as the free semilattice in the category. In contrast to the other type constructors, the powerdomain does not preserve the subcategory of cpos. Indeed we show that the powerdomain has interesting computational properties that differ from those of the usual convex powerdomain on cpos. We end by considering the solution of recursive domain equations. The surprise here is that the limit-colimit coincidence fails. Nevertheless, ...

We define a proof system `a la Hoare for a common kernel of existing data-- parallel languages. This kernel includes conditioning constructs and non local control transfer such as data--parallel break and continue. Assertions are usual predicates, where extent of parallelism is to be described by specific variables. We show that it is moreover possible to define a weakest (liberal) preconditions calculus, and to deduce the relative completeness of the proof system. Our main contribution is to propose a simple substitution mechanism as a basic tool to express complex manipulations of the extent of parallelism, as found in recent high--level data--parallel languages. It enables us to reuse the classical assertional setting of Hoares's Logic, with expected properties, in the framework of data--parallel languages. Keywords : Concurrent Programming; Specifying and Verifying and Reasoning about Programs; Semantics of Programming Languages; Data-Parallel Languages; Proof System; Hoare Logic;...

vate communication, as cited in Stix, 1973. Carreras, B. A., Lynch, V. E., and Garcia, L. (1991). Electron Diamagnetic Effects on the Resistive Pressure-Gradient-Driven Turbulence and Flow 167 168 Bibliography Generation. Phys. Fluids B 3, 1438. Catto, P. J. and Tsang, K. T. (1978). Trapped Electron Instability in Tokamaks: Analytic Solution of the Two-Dimensional Eigenvalue Problem. Phys. Fluids 21, 1381. Cheng, C. Z. and Chen, L. (1981). Ballooning-Mode Theory of TrappedElectron Instabilities in Tokamaks. Nucl. Fusion 21, 403. Chew, G. F., Goldberger, M. L., and Low, F. E. (1956). Boltzmann Equation and the One-Fluid Hydrodynamic Equations in the Absence of Particle Collisions. Proc. Roy. Soc. A236, 112. Cohen, B.,