this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.

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

We propose a definition by reducibility of sequentiality for the interpretations of higher-order programs and prove the equivalence between this notion and strong stability.

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

We consider sets CtR(σ) of total, continuous functionals of type σ over the reals. A subset A ⊆ CtR(σ) is reducible if A can be reduced to totality in one of the other spaces. We show that all Polish spaces are homeomorphic to a reducible subset of R → R and that the class of reducible sets is closed under the formation of function spaces and some comprehension. 1

Most of the structures one deals with in Computer Science have a discrete and effective character: (finite) graphs, nets, trees in programming and formal languages, algorithms over finite strings or natural numbers, circuits etc... By continuous structures we mean "smooth" spaces, usually of cardinality not less than continuum, where interesting topological or order properties give some information on, say, classes of functions over them. By discussing results from various areas of Mathematical Computer Science, we stress the role of continuous structures as tools for proving results about discrete or even finite structures. In particular we overview results concerning functionals in computability theory, trees in lambda calculus, boolean circuits in complexity theory and relate the finitary/combinatorial nature of the problems with their continuous solutions. We mostly focus on the methodology, and just hint to the technical aspects of the results presented. 2.1 Introduction In order...

In this paper we develop an approach to the notion of computable functionals in a very abstract setting not restricted to Turing or, say, polynomial computability. We assume to start from some basic class of domains and a basic class of functions defined on these domains. (An example may be natural numbers with polytime computable functions). Then we define what are "all" corresponding functionals of higher types which add nothing new to these basic functions. We call such functionals computable or, more technically and adequately speaking, substitutable. (Similarly, in D.Scott's domains we say about continuous functionals as about far-reaching abstraction of computable ones.) Our results are applicable to quite arbitrary (complexity) classes of functions, satisfying a very general Assumption. 1 Introduction The problem raised up in this paper is related to the general notions of computations in different programming paradigms over different data types. The problem may be ...

We sketch a constructive formal theory TCF + of computable functionals which allows to reason not only about the functionals themselves but also about their finite approximations. Types are built from base types by the formation of function types, ρ → σ. The intended semantical domains for the base types are non-flat free algebras, given by their constructors, where the latter are injective and have disjoint ranges; both properties do not hold in the flat case. In this setting we give an informal proof (based on Berger [2]) of Kreisel’s density theorem [7], and an adaption of Plotkin’s definability theorem [10, 11]. We then show that both proofs can be formalized in TCF +. The naive model of a finitely typed theory like TCF + is the full set theoretic hierarchy of functionals of finite types. However, this immediately leads to higher cardinalities, and does not lend itself well for a constructive theory of computability. A more appropriate semantics for typed languages has its roots in work of Kreisel [7] (which used formal neighborhoods) and Kleene [6]. This line of research was developed in a mathematically more