### A General Form of Relative Recursion

Jaap van Oosten
Source: Notre Dame J. Formal Logic Volume 47, Number 3 (2006), 311-318.

#### Abstract

The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.

First Page:
Primary Subjects: 03B40
Secondary Subjects: 68N18
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1163775438
Digital Object Identifier: doi:10.1305/ndjfl/1163775438
Mathematical Reviews number (MathSciNet): MR2264700
Zentralblatt MATH identifier: 1113.03014

### References

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Mathematical Reviews (MathSciNet): MR1981211
Zentralblatt MATH: 1046.03038
Digital Object Identifier: doi:10.1017/S0305004102006424
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Mathematical Reviews (MathSciNet): MR1981211
Digital Object Identifier: doi:10.1017/S0305004102006424
Zentralblatt MATH: 1046.03038
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