this paper we show that the two approaches are equivalent for a

. A new treatment of data refinement in typed lambda calculus is proposed, based on pre-logical relations [HS99] rather than logical relations as in [Ten94], and incorporating a constructive element. Constructive data refinement is shown to have desirable properties, and a substantial example of refinement is presented. 1 Introduction Various treatments of data refinement in the context of typed lambda calculus, beginning with Tennent's in [Ten94], have used logical relations to formalize the intuitive notion of refinement. This work has its roots in [Hoa72], which proposes that the correctness of a concrete version of an abstract program be verified using an invariant on the domain of concrete values together with a function mapping concrete values (that satisfy the invariant) to abstract values. In algebraic terms, what is required is a homomorphism from a subalgebra of the concrete algebra to the abstract algebra. A strictly more general method is to take a homomorphic relatio...

Abstract. We compare the definability of total functionals over the reals in two functional-programming approaches to exact real-number datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to second-order types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at third-order types. However, we relate this question to a purely topological conjecture about the Kleene-Kreisel continuous functionals over the natural numbers. Finally, although it is known that, in the extensional approach, parallel primitives are necessary for programming total first-order functions, we demonstrate that, in the intensional approach, such primitives are not needed for second-order types and below. 1

In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem of when a topological space has a λ-admissible and κ-based domain representation. We also prove that there is a natural cartesian closed category of countably based and countably admissible domain representations. These results are generalisations of [Sch02]. 1

In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the Kleene-Kreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zero-dimensional for each type. As a tool of independent interest, we will construct topological embeddings of the Kleene-Kreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types. 1

There are two main approaches to obtaining "topological" cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed --- for example, the category of sequential spaces. Under the other, one generalises the notion of space --- for example, to Scott's notion of equilogical space. In this paper we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably-based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. This category consists of certain "!-projecting" topological quotients of countably-based topological spaces, and contains, in particular, all countably-based spaces. We show that this category is cartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the cate...

We show that the continuous existential quantifier is not definable in Escard6's Real-PCF from all functionals equivalent to a given total one in a uniform way.

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.

The Kleene-Kreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbers as inputs and finally survey some recent applications of this generalization. 1

Abstract. General methods of investigating effectivity on regular Hausdorff (T3) spaces is considered. It is shown that there exists a functor from a category of T3 spaces into a category of domain representations. Using this functor one may look at the subcategory of effective domain representations to get an effectivity theory for T3 spaces. However, this approach seems to be beset by some problems. Instead, a new approach to introducing effectivity to T3 spaces is given. The construction uses effective retractions on effective Scott–Ershov domains. The benefit of the approach is that the numbering of the basis and the numbering of the elements are derived at once. 1