5 research outputs found
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
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
Fixed point theorems for precomplete numberings
Contains fulltext :
205967.pdf (preprint version ) (Open Access