Total sets and objects in domain theory

Berger, Ulrich

doi:10.1016/0168-0072(93)90038-f

Cited by 69 publications

(142 citation statements)

References 11 publications

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“…For part (ii), it suffices to note that the constructions in the above proof are effective, so that the realizers α k , β k may all be chosen from K eff 2 . Part (iii) is proved in (Berger 1993) in the form we require. The key ingredients of a proof of (iii) will also appear in the proof of Proposition 8.4 below.…”

Section: Realizations Of Type Structuresmentioning

confidence: 99%

“…Part (iii), which corresponds to a strengthened version of the generalized KLS theorem, appears in (Berger 1993). Parts (i) and (ii) do not seem to have appeared explicitly in the literature, although a closely related fact, with L in place of P, can be read off from the results of (Bauer 2000;Bauer 2002).…”

Section: Realizations Of Type Structuresmentioning

confidence: 99%

“…The generalization to higher types was noted in (Kreisel 1959); the first complete proofs appeared in (Hyland 1975;Ershov 1976a). The treatment in (Berger 1993) is particularly well adapted to our present purposes.…”

Section: Thusmentioning

confidence: 99%

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On the ubiquity of certain total type structures

Longley¹

2007

Math. Struct. Comp. Sci.

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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.

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Section: Realizations Of Type Structuresmentioning

confidence: 99%

Section: Realizations Of Type Structuresmentioning

confidence: 99%

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On the ubiquity of certain total type structures

Longley¹

2007

Math. Struct. Comp. Sci.

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“…It is known, [Ber93], that the following generalization of the "KLS Theorem" holds: for any continuous f as in the diagram, there exist g and h making the squares commute. (This is a non-effective version of the "KLS Theorem" in [Ber93]. The proof uses the density of the inclusion T real1 ⊆ real 1 .)…”

Section: Extensionalizationmentioning

confidence: 99%

“…Following [Ber93], we define the set of total elements T τ ⊆ τ , where τ is the domain interpreting a PCF type τ . This is by:…”

Section: Extensionalizationmentioning

confidence: 99%

Comparing Functional Paradigms for Exact Real-Number Computation

Bauer

Escardó

Simpson

2002

Automata, Languages and Programming

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Abstract. We compare the definability of total functionals over the reals in two functional-programming approaches to exact real-number computation: the extensional approach, in which one has an abstract 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.

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A Relationship between Equilogical Spaces and Type Two Effectivity

Bauer

2002

MLQ - Math. Log. Quart.

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In this paper I compare two well studied approaches to topological semanticsthe domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related.First, we show that Rep(B ) is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep(B ) and Equ. The inclusion Rep(B ) → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep(B ) and Equ are essentially different. However, in a second comparison we show that Rep(B ) and Equ do share a common cartesian closed subcategory that contains all countably based T0-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another.

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Total sets and objects in domain theory

Cited by 69 publications

References 11 publications

On the ubiquity of certain total type structures

On the ubiquity of certain total type structures

Comparing Functional Paradigms for Exact Real-Number Computation

A Relationship between Equilogical Spaces and Type Two Effectivity

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