We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

this paper we study some extensions of the Kleene-Kreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functionals as the total elements in a hierarchy of Ershov-Scott-domains of partial continuous functionals. In this setting the density theorem says that the total functionals are topologically dense in the partial ones, i.e. every nite (compact) functional has a total extension. We will extend this theorem from function spaces to dependent products and sums and universes. The key to the proof is the introduction of a suitable notion of density and associated with it a notion of co-density for dependent domains with totality. We show that the universe obtained by closing a given family of basic domains with totality under some quantiers has a dense and co-dense totality provided the totalities on the basic domains are dense and co-dense and the quantiers preserve density and co-density. In particular we can show that the quantiers and have this preservation property and hence, for example, the closure of the integers and the booleans (which are dense and co-dense) under and has a dense and co-dense totality. We also discuss extensions of the density theorem to iterated universes, i.e. universes closed under universe operators. From our results we derive a dependent continuous choice principle and a simple order-theoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the Kreisel-Lacombe-Shoeneld-Theorem showing the coincidence of the hereditarily eectively continuous hierarchy...

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 present a realizability interpretation of co-inductive types based on partial equivalence relations (per's). We extract from the per's interpretation sound rules to type recursive definitions. These recursive definitions are needed to introduce "infinite" and "total" objects of co-inductive type such as an infinite stream or a non-terminating process. We show that the proposed type system enjoys the basic syntactic properties of subject reduction and strong normalization with respect to a confluent rewriting system first studied by Gimenez. We also compare the proposed type system with those studied by Coquand and Gimenez. In particular, we provide a semantic reconstruction of Gimenez's system which suggests a rule to type nested recursive definitions.

Using a categorical model of abstract data types [2, 3, 13, 15], we show, following the Kozen's technique [10] and Tarjan's constructions for a deterministic automaton [16], that if two unambiguous regular expressions dene the same regular language, then they represent two isomorphic abstract data types. 1