this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of higher-type functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as well---and that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the so-called untyped -calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary way---avoiding heavy prerequisites (such as category theory) but still maintaining some level of abstraction---with the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...

We show the equivalence of several different axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a finite number of possibilities with arbitrary probabilities, whereas the poorest one has binary choice with equal probabilities as the only operation. The remaining theories lie in between; one of them is the theory of binary choice by Graham [1]. 1 Introduction A probabilistic programming language could contain different kinds of language constructs to express probabilistic choice. In a rather poor language, there might be a construct x \Phi y, whose semantics is a choice between the two possibilities x and y with equal probabilities 1=2. The `possibilities' x and y can be statements in an imperative language or expressions in a functional language. A quite rich language could contain a construct [p 1 : x 1 ; : : : ; p n : x n ],...

. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of pre-orders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...

Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different condition---that of coherence---which has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a cartesian closed category which has fixed points for domain equations. It is shown that a "universal domain" exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and applied to other classes of domains. Universal domains constructed in this fashion enjoy an additional property: they are saturated. We show that there is exactly one such domain in each of the classes under consideration. 1 Introduction. The first structures used as a mathematical foundation for the denotational semantic...

We demonstrate how order-theoretic abstractions can be useful in identifying, formalizing, and exploiting relationships between seemingly dissimilar AI algorithms that perform computations on partially-ordered sets. In particular, we show how the order-theoretic concept of an anti-chain can be used to provide an efficient representation for such sets when they satisfy certain special properties. We use anti-chains to identify and analyze the basic operations and representation optimizations in the version space learning algorithm [10] and the assumption-based truth maintenance system (ATMS) [2, 3]. Our analysis allows us to (1) extend the known theory [7, 10, 8] of admissibility of concept spaces for incremental version space merging, and (2) develop new, simpler label-update algorithms for ATMS's with DNF assumption formulas. Contents 1 Introduction 2 2 Representing Sets as Anti-Chains 4 3 Version Spaces 17 4 Assumption-Based Truth Maintenance Systems 32 5 Extended ATMS's 46 6 Ackno...

. Disjunctive systems are a representation of L-domains. They use sequents of the form X ` Y , with X finite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an Ldomain. On the other hand, via the notion of stable neighborhoods, every L-domain can be represented as a disjunctive system. More generally, we have a categorical equivalence between the category of disjunctive systems and the category of L-domains. A natural classification of domains is obtained in terms of the style of the entailment: when jXj = 2 and jY j = 0 disjunctive systems determine coherent spaces; when jY j 1 they represent Scott domains; when either jXj = 1 or jY j = 0 the associated cpos are distributive Scott domains; and finally, without any restriction, disjunctive systems give rise to L-domains. 1 Introduction Discovered by Coquand [Co90] and Jung [Ju90] independently, L-domains form one of the maximal cartesian closed categories of algebraic cpos. Tog...

The paper addresses the question of when the least-fixed-point operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a sufficient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold. The condition is quite mild, and the least-fixed-point operator is so characterised in many of the most commonly used categories of domains. By using retractions, the characterisation extends to the associated cartesian closed categories of continuous cpos. However, dinaturality does not always characterise the least-fixed-point operator. We show that in cartesian closed full subcategories of the category of continuous lattices the characterisation fails. 1 Introduction Mulry [7] showed that, under general conditions on a category of domains, the least-fixed-point operator, lfp D : D D ! D, is a dinatural transformation ...

\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an I-category as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete I-categories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...

We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections with sup P = id D . They carry a canonical uniformity and thus a topology. We relate their properties such as completeness and compactness to properties of the poset and the projection set. We show that each monotone net in D is convergent if and only if (D; ) is an algebraic domain such that the images of the projections are precisely the compact elements of (D; ). We call these domains P-domains and characterize them as inverse limits of posets satisfying the ascending chain condition. Moreover, we describe P-domains by a certain system of so-called "complete" subsets. We prove that if the set of compact elements of an algebraic domain is mub-complete, then it is a P-domain if and only if the mub-closure of every finite set of compact elements fulfils the ascending chain condition. Furthermore, we characte...

This paper shows that Axiom d and Axiom I are important when one works within the realm of Scott-domains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scott-domain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, for any finitary Scott-domain D. Therefore, the category of dI-domains is the largest cartesian closed category within omega-algebraic, bounded complete domains, with the exponential being the stable function space. 1 Introduction Among Scott's many insights which shaped the whole area of domain theory, one is that the partial ordering of a domain should be interpreted as the ordering about information. "Thus," wrote Scott [16], "x v y means that