Abstract not found

Abstract not found

It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene-Kreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.

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

It is well-known that stable models (as dI-domains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cannot be defined by programs ofPCF. In this paper, a paradigmatic programming language namedStPCF is proposed, which extends the languagePCF with two additional operators. The operational description of the extended language is presented in an effective way, although the evaluation of one of the new operators cannot be formalized in a PCF-like rewrite system. SinceStPCF can define all finite cliques of coherence spaces the above gap with stable models is filled, consequently stable models are fully abstract for the extended language. 1