Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract

Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

Uncountable Limits and the Lambda Calculus

In this paper we address the problem of solving recursive domain equations using uncountable limits of domains. These arise for instance, when dealing with the omega_1-continuous function-space constructor and are used in the denotational semantics of programming languages which feature unbounded choice constructs. Surprisingly, the category of cpo’s and omega_1-continuous embeddings is not omega_0-cocomplete. Hence the standard technique for solving reflexive domain equations fails. We give two alternative methods. We discuss also the issue of completeness of the lambda beta eta-calculus w.r.t reflexive domain models. We show that among the reflexive domain models in the category of cpo’s and omega_0-continuous functions there is one which has a minimal theory. We give a reflexive domain model in the category of cpo’s and omega_1-continuous functions whose theory is precisely the lambda beta eta theory. So omega_1-continuous lambda-models are complete for the lambda beta eta-calculus

Semantic Domains for Combining Probability and Non-Determinism

AbstractWe present domain-theoretic models that support both probabilistic and nondeterministic choice. In [A. McIver and C. Morgan. Partial correctness for probablistic demonic programs. Theoretical Computer Science, 266:513–541, 2001], Morgan and McIver developed an ad hoc semantics for a simple imperative language with both probabilistic and nondeterministic choice operators over a discrete state space, using domain-theoretic tools. We present a model also using domain theory in the sense of D.S. Scott (see e.g. [G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. Continuous Lattices and Domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003]), but built over quite general continuous domains instead of discrete state spaces.Our construction combines the well-known domains modelling nondeterminism – the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science, pages 186–195. IEEE Computer Society Press, 1989] modelling probabilistic choice. The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4). In order to prove the desired universal equational properties of these combined powerdomains, we develop sandwich and separation theorems of Hahn-Banach type for Scott-continuous linear, sub- and superlinear functionals on continuous directed complete partially ordered cones, endowed with their Scott topologies, in analogy to the corresponding theorems for topological vector spaces in functional analysis (see Chapter 3). In the end, we show that our semantic domains work well for the language used by Morgan and McIver