A novel upper power domain construction is defined by means of strongly compact sets. Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk. The notion of strong compactness allows a proof of stronger properties than compactness would, e.g. an intrinsic universal property of the upper power construction, and its commutation with the lower construction.

Following the program of Moggi, the semantics of a simple non-deterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle non-termination properly. Instead, a novel construction is proposed and investigated. It embodies both nondeterminism (choice and failure) and possible non-termination caused by recursion. 1 Introduction Following the proposals of Moggi [Mog89, Mog91b], functional languages with various notions of computations can be denotationally described by means of monads. Monads are constructions mapping domains of values into domains of computations for these values. Computations involving destructive assignments, for instance, are handled by the state transformer monad [Wad90], whereas computations by non-deterministic choice are handled by power domain constructions [Plo76, Smy78, Gun90]. All known power domain constructions and many others ar...

If a strong monad M is used to define the denotational semantics of a functional language with computations, a product operation \Gamma \Theta : MX \Theta MY !M (X \Theta Y ) is needed to define the semantics of pairing. Every strong monad is equipped with two standard products, which correspond to left-to-right and right-to-left evaluation. We study the algebraic properties of these standard products in general. Then we define alternative products with similar properties for strict and parallel evaluation in the special case of strong monads in DCPO which are obtained as free constructions w.r.t. various theories of non-deterministic computation, including both classical and probabilistic theories. 1 Introduction Following the proposals of Moggi [12, 13], functional languages with various notions of computations can be denotationally described by means of (particular kinds of) monads. Informally, these monads are constructions mapping domains of values into domains of computations...

We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss [0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing `+' with `' in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if O(X) is the Scott-topology of a continuous domain. The notion of [0; 1]- and [0; 1]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain Poss(D) is the free [0; 1]-module and Poss [0;1] (D) the free [0; 1]-module over a continuous domain D. 1 Possibility Measures This extended abstract attempts to recast som...

This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...

We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by Boehm and Cartwright, is to use a non-deterministic test. For any two rational numbers p < q and any real number x, at least one of the relations p < x or x < q can be determined to hold; thus, an operator rtest is used, whose evaluation never diverges when x is a real number: 1. rtestp,q(x) evaluates to true or to false, 2. rtestp,q(x) may evaluate to true iff x < q and 3. rtestp,q(x) may evaluate to false iff p < x. Since a program can in general produce different results in different runs, Escardó and Marcial-Romero took the view in previous work that programs of real-number type denote sets of real numbers, and the question arose as to which power domains would be suitable for modelling the behaviour of rtest. It was shown that, among the known power domains,