. We introduce a domain-theoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introduce a natural declarative semantics and a fixed-point semantics for disjunctive logic programs, and prove their equivalence; finally, we apply our results to give both a syntax and semantics for default logic in any coherent algebraic dcpo. 1. Introduction Domain theory, as introduced by Scott in the 1970's, has many connections with logic. Such connections are usually made by extracting an appropriate language /syntax from a category of domains. To name a few examples, we have Abramsky's "domain theory in logical form" [Abr91], Scott's own representation of Scott domains as information systems [Sco82], extended to other domains by Zhang [Zha91], and Smyth's treatment of observable prope...

Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstraction, and we put a few old and recent results of this kind in perspective.

This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.

A longstanding open problem in lambda-calculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambda-calculus whose theory is exactly the beta-theory or the beta-eta-theory. A related question, raised recently by C.Berline, is whether, given a class of lambda-models, there is a minimal equational theory represented by it.

We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos. We show how to identify the notion of formal concept in the domain theoretic setting. In particular, we show that a special instance of the resolution rule from the domain logic coincides with the concept closure operator from formal concept analysis. The results shed light on the use of contexts and domains for knowledge representation and reasoning purposes.

We introduce a semantical model based on operator algebras and we show the suitability of this model to capture both a quantitative version of non-determinism (in the form of a probabilistic choice) and concurrency. We present the model by referring to a generic language which generalises various probabilistic concurrent languages from different programming paradigms. We discuss the relation between concurrency and the commutativity of the resulting semantical domain. In particular, we use Gelfand's representation theorem to relate the semantical models of synchronisation-free and fully concurrent versions of the language. A central aspect of the model we present is that it allows for a unified view of both operational and denotational semantics for a concurrent language.

After Scott, mathematical models of the type-free lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any non-trivially partially ordered model (order-incompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is order-incomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the order-incompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one -term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).

This paper proposes a representation for plans and actions based on the algebraic theory of processes. It is argued that the requirements of plan-execution are better met by representing actions through the processes by which changes occur than by the more widely used state-change representation. A simple algebra of plans, based on process-combinators, is described and shown to be adequate for a wide variety of plans. The implications of this type of plan-representation are discussed and its advantages for meta-reasoning (including plan-comparison) outlined. This paper presents the theory of processes from the point of view of planning and describes a novel method of plan-comparison which draws upon ideas in domain theory. Keywords: Plan-representation; plan-execution; plan-comparison; philosophical foundations A note on processes for plan-execution and powerdomains for plan-comparison David Pym Louise Pryor David Murphy Queen Mary & Westfield College Department of Artificial Intell...

W.C. Rounds and G.-Q. Zhang have recently proposed to study a form of resolution on algebraic domains [1]. This framework allows reasoning with knowledge which is hierarchically structured and forms a (suitable) domain, more precisely, a coherent algebraic cpo as studied in domain theory. In this paper, we give conditions under which a resolution theorem --- in a form underlying resolution-based logic programming systems --- can be obtained. The investigations bear potential for engineering new knowledge representation and reasoning systems on a firm domain-theoretic background.

In this collection we try to give anoverview of some selected topics in Domain Theory and Denotational Semantics. In doing so, we rst survey the mathematical universes which have been used as semantic domains. The emphasis is on those ordered structures which have beenintroduced by Dana Scott in 1969 and which gure under the name (Scott-) domains. After surveying developments in the concrete theory of domains we describe two newer developments, the axiomatic and the synthetic approach. In the second part we look at three computational phenomena in detail, namely, sequential computation, polymorphism, and mutable state, and at the challenges that these pose for a mathematical model. This presentation does by no means exhaust the various approaches to denotational semantics and it certainly does not describe all possible mathematical techniques which havebeen used to describe various aspects of programs. We hope that, nevertheless, it illustrates how a particular challenge (namely the modelling of recursive de nitions) has given rise to an immensely rich theory, both in its general parts and in its applications.