Abstract
For any representation δ there is a derived numbering v δ, defined by ν δ(i):=δφi, of the δ-computable elements. In this chapter the relation between δ-computability and ν δ-computability is investigated. It is easy to show that the restriction of a (δ,δ,)-computable function to the computable elements is (ν δ,ν δ’)-computable. The converse does not seem to be true in general. Only for two special cases a positive answer is known: for “effective” metric spaces (special cases have been proved independently by Ceitin, by Kreisel, Lacombe, and Shoenfield, and by Moschovakis) and for “computable” cpo’s (a special case has been proved by Myhill and Sheperdson). As a corollary we obtain a theorem which characterizes the ν δ-r.e. subsets of the δ-computable elements for “computable” cpo’s (Rice /Shapiro theorem). The proofs of the main theorems are rather sophisticated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Weihrauch, K. (1987). Type 1 Computability Type 2 Computability. In: Computability. EATCS Monographs on Theoretical Computer Science, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69965-8_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-69965-8_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-69967-2
Online ISBN: 978-3-642-69965-8
eBook Packages: Springer Book Archive