This special issue of Mathematical Structures in Computer Science is devoted to the Proceedings of the International Workshop Logic, Domains, and Programming Languages that took place from May 24 to 27, 1995, in Darmstadt, Germany.

We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a Gδ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

Algebraic CPOs naturally generalize to finitely accessible categories, and Scott domains (i.e., consistently complete algebraic CPOs) then correspond to what we call Scott-complete categories: finitely accessible, consistently (co-)complete categories. We prove that the category SCC of all Scott-complete categories and all continuous functors is cartesian closed and provides fixed points for a large collection of endofunctors. Thus, SCC can serve as a basis for semantics of computer languages.

The purpose of this note is to show that the correctness of a multiplicative proof net with mix is equivalent to its semantic correctness: a proof structure is a proof net if and only if its semantic interpretation is a clique, where one given finite coherence space interprets all propositional variables.

This is just an example of what can be done with these kinds of semantic techniques; for more information and further results, the reader is referred to Retoré (1994).

We introduce the notions of premonoidal category and premonoidal functor, and show how these can be used in the denotational semantics of programming languages. We characterize the semantic definitions of Eugenio Moggi's monads as notions of computation, exhibit a representation theorem for our premonoidal setting in terms of monads, and give a fibrational setting for the structure.

We propose a uniform way of isolating a subcategory of predomains within the category of modest sets determined by a partial combinatory algebra (PCA). Given a divergence on a PCA (which determines a notion of partiality), we identify a candidate category of predomains, the well-complete objects. We show that, whenever a single strong completeness axiom holds, the category satisfies appropriate closure properties. We consider a range of examples of PCAs with associated divergences and show that in each case the axiom does hold. These examples encompass models allowing a ‘parallel’ style of computation (for example, by interleaving), as well as models that seemingly allow only ‘sequential’ computation, such as those based on term-models for the lambda-calculus. Thus, our approach provides a uniform approach to domain theory across a wide class of realizability models. We compare our treatment with previous approaches to domain theory in realizability models. It appears that no other approach applies across such a wide range of models.

We formalize a partial evaluator for Eugenio Moggi's computational metalanguage. This formalization gives an evaluation-order independent view of binding-time analysis and program specialization, including a proper treatment of call unfolding. It also enables us to express the essence of ‘control-based binding-time improvements’ for let expressions. Specifically, we prove that the binding-time improvements given by ‘continuation-based specialization’ can be expressed in the metalanguage via monadic laws.

This paper shows that it is precisely the complete metrizable separable metric spaces that can be realized as the set of maximal points of an ω-continuous dcpo, where the set of maximal points is topologized with the relative Scott topology.

This paper develops the functional part of a theory of action semantics for reasoning about programs. Action notation, the specification language of action semantics, is given an evaluation semantics, and operational techniques from process theory and functional programming are applied in the development of a versatile action theory. The power of the theory is demonstrated by means of action semantic proofs of functional program equivalences.

Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. In this paper we prove an enrichment theorem showing that every axiomatic domain-theoretic category can be endowed with an intensional notion of approximation, the path relation, with respect to which the category Cpo-enriches.

Our analysis suggests more liberal notions of domains. In particular, we present a category where the path order is not ω-complete, but in which the constructions of domain theory (such as, for example, the existence of uniform fixed-point operators and the solution of domain equations) are available.