Model of Partial Continuous Functionals

Ershov, Yu. L.

doi:10.1016/s0049-237x(09)70439-2

Cited by 52 publications

(52 citation statements)

References 4 publications

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“…The model HEO of hereditarily effective operations, which is defined similarly to 2., but with natural numbers and partial recursive function application instead of terms and term application [20]. 4. The model C of continuous functionals, which we define following the same pattern as in 2. and 3.…”

Section: Modelsmentioning

confidence: 99%

Classical truth in higher types

Berger

2008

Mathematical Logic Qtrly

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We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical set-theoretic finite type hierarchy and the constructively more meaningful models of Continuous Functionals, Hereditarily Effective Operations, as well as the closed term model of Gödel's system T . The main results are characterisations of prenex classes for which truth in the full set-theoretic model transfers to truth in the other models. As a corollary we obtain that the axiom of choice is not conservative over Gödel's system T with classical logic for closed ∃ 2 -formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective.

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Section: Modelsmentioning

confidence: 99%

Classical truth in higher types

Berger

2008

Mathematical Logic Qtrly

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“…The sets \D P \ of partial continuous functionals of type p are the proper domains for computable functionals (Kreisel in [7] and Ersov in [3] give convincing arguments for this) and also for the partial primitive recursive functionals we want to study here. In Section 2 we will give a definition of the sets \D P \, in a form convenient for our later arguments.…”

Section: Finite Functionalsmentioning

confidence: 99%

“…Of special importance (and much studied in the literature, e.g. in [13], [3], [8]) are the computable ones, which by definition are those given by recursively enumerable ideals of finite functionals (in \Dp-> a \).…”

Section: Primitive Recursionmentioning

confidence: 99%

“…We investigate Gödel's notion of a primitive recursive functional of higher type [5] in the context of partial continuous functional as introduced by Kreisel in [7] and developed mainly by Scott (see [13], [3]). To make this paper readable for people not familiar with the theory of partial continuous functionals we have included a short exposition of the basic material, in a form convenient for our later arguments.…”

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confidence: 99%

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Primitive Recursion on the Partial Continuous Functionals

Schwichtenberg¹

1991

Informatik Und Mathematik

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“…Scott domains (Scott 1970) provide denotational semantics for a wide range of programming languages and carry a natural notion of computability (Ershov 1977). Using domain representations (Blanck 2000;Stoltenberg-Hansen and Tucker 1995;Stoltenberg-Hansen and Tucker 2008), the domain-theoretic notion of computability can be extended to a large class of topological spaces, and, moreover, important classes of topological spaces can be characterised by the kind of domain representations they admit (Hamrin 2005).…”

Section: Introductionmentioning

confidence: 99%

Domain representations of spaces of compact subsets

Berger¹,

Blanck²,

Køber³

2010

Math. Struct. Comp. Sci.

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We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

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Model of Partial Continuous Functionals

Cited by 52 publications

References 4 publications

Classical truth in higher types

Classical truth in higher types

Primitive Recursion on the Partial Continuous Functionals

Domain representations of spaces of compact subsets

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