We show that to every recursive total continuous functional there is a representative of in the hierearchy of partial continuous funcriohals such that is S1 - S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative will be PCF-definable over the partial continuous functionals, where PCF is Plotkin's programming language for computable functionals.

. We show that dependent sums and dependent products of continuous parametrizations on domains with dense, codense, and natural totalities agree with dependent sums and dependent products in equilogical spaces, and thus also in the realizability topos RT(P!). Keywords: continuous functionals, dependent type theory, domain theory, equilogical spaces. 1 Introduction Recently there has been a lot of interest in understanding notions of totality for domains [3, 23, 4, 18, 21]. There are several reasons for this. Totality is the semantic analogue of termination, and one is naturally interested in understanding not only termination properties of programs but also how notions of program equivalence depend on assumptions regarding termination [21]. Another reason for studying totality on domains is to obtain generalizations of the nite-type hierarchy of total continuous functionals by Kleene and Kreisel [11], see [8] and [19] for good accounts of this subject. Ershov [7] showed how the Klee...

ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses then they are totally equivalent operationally iff they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott's model of continuous functions, Milner's fully abstract model of PCF and their effective submodels. We investigate how totality differs between these models. Some apparently rather difficult open problems arise, essentially concerning whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivale...

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 give a characterisation of an extension of the Kleene-Kreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.