Domain interpretations of martin-löf’s partial type theory

Palmgren, Erik; Stoltenberg-Hansen, Viggo

doi:10.1016/0168-0072(90)90044-3

Cited by 35 publications

(39 citation statements)

References 6 publications

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“…First, we will recall some of the fundamental domain theory, mainly based on [11], and introduce the notion of parametrization from [9]. We also present some basic facts of p -spaces, as well as a standard domain representation of the reals.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [9], effectivity of parametrizations is studied. Following their convention, we say that a parametriza- The general notions of computable and effective structures given in [11] are applied in [2] to define computable and effective metric spaces.…”

Section: Effective Parametrizationsmentioning

confidence: 99%

“…Applying this to the parametrization F L , the uniform selections from { p } p∈P will be defined as those functions over P represented in the dependent product of domains. The elements of Π(D, F ) were introduced as p-continuous functions over D in [9], and we will adopt this notion for selection functions over P .…”

Section: Dependent Type Constructionsmentioning

confidence: 99%

“…In this context the notion of parametrization has arisen, formalizing the concept of a family of domains uniformly parametrized over a domain. Parametrizations of domains were carefully studied by Palmgren and Stoltenberg-Hansen in [9].…”

Section: Introductionmentioning

confidence: 99%

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Uniform domain representations of ℓ^p‐spaces

Køber

2007

Mathematical Logic Qtrly

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The category of Scott-domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach-space is representable over a separable domain. A large class of topological spaces, including all Banach-spaces, is representable by domains, and in domain theory, there is a well-understood notion of parametrizations over a domain. We explore the link with parameter-dependent collections of spaces in e. g. functional analysis through a case study of p -spaces. We show that a well-known domain representation of p as a metric space can be made uniform in the sense of parametrizations of domains. The uniform representations admit lifting of continuous functions and are effective in p. Dependent type constructions apply, and through the study of the sum and product spaces, we clarify the notions of uniformity and uniform computability.

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Section: Preliminariesmentioning

confidence: 99%

Section: Effective Parametrizationsmentioning

confidence: 99%

Section: Dependent Type Constructionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniform domain representations of ℓ^p‐spaces

Køber

2007

Mathematical Logic Qtrly

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“…This was exploited in [34] where Ershov gave a domain representation of the Kleene-Kreisel continuous functionals. An effective and adequate domain model of Martin-Löf partial type theory is given in Palmgren and Stoltenberg-Hansen [57] which has been extended in Waagbø [84] to provide a domain representation of Martin-Löf total type theory (see also Berger [6] and Normann [54]). …”

Section: Domain Representation Theorymentioning

confidence: 99%

Computability on Topological Spaces via Domain Representations

Stoltenberg-Hansen

Tucker

2008

New Computational Paradigms

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Domains are ordered structures designed to model computation with approximations. We give an introduction to the theory of computability for topological spaces based upon representing topological spaces and algebras using domains. Among the topics covered are: different approaches to computability on topological spaces; orderings, approximations and domains; making domain representations; effective domains; classifying representations; type two effectivity and domains; special representations for inverse limits, regular spaces and metric spaces. Lastly, we sketch a variety of applications of the theory in algebra, calculus, graphics and hardware.

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