71 research outputs found
The Rice-Shapiro theorem in Computable Topology
We provide requirements on effectively enumerable topological spaces which
guarantee that the Rice-Shapiro theorem holds for the computable elements of
these spaces. We show that the relaxation of these requirements leads to the
classes of effectively enumerable topological spaces where the Rice-Shapiro
theorem does not hold. We propose two constructions that generate effectively
enumerable topological spaces with particular properties from wn--families and
computable trees without computable infinite paths. Using them we propose
examples that give a flavor of this class
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in “A Workshop on Realizability Semantics and Its Applications”, Trento, Italy, June 30–July 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Fixpoints and relative precompleteness
We study relative precompleteness in the context of the theory of numberings,
and relate this to a notion of lowness. We introduce a notion of divisibility
for numberings, and use it to show that for the class of divisible numberings,
lowness and relative precompleteness coincide with being computable.
We also study the complexity of Skolem functions arising from Arslanov's
completeness criterion with parameters. We show that for suitably divisible
numberings, these Skolem functions have the maximal possible Turing degree. In
particular this holds for the standard numberings of the partial computable
functions and the c.e. sets.Comment: 12 page
Generalizations of the Recursion Theorem
We consider two generalizations of the recursion theorem, namely Visser's ADN
theorem and Arslanov's completeness criterion, and we prove a joint
generalization of these theorems
Computability in partial combinatory algebras
We prove a number of elementary facts about computability in partial
combinatory algebras (pca's). We disprove a suggestion made by Kreisel about
using Friedberg numberings to construct extensional pca's. We then discuss
separability and elements without total extensions. We relate this to Ershov's
notion of precompleteness, and we show that precomplete numberings are not 1-1
in general
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
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