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2006 A General Form of Relative Recursion
Jaap van Oosten
Notre Dame J. Formal Logic 47(3): 311-318 (2006). DOI: 10.1305/ndjfl/1163775438


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.


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Jaap van Oosten. "A General Form of Relative Recursion." Notre Dame J. Formal Logic 47 (3) 311 - 318, 2006.


Published: 2006
First available in Project Euclid: 17 November 2006

zbMATH: 1113.03014
MathSciNet: MR2264700
Digital Object Identifier: 10.1305/ndjfl/1163775438

Primary: 03B40
Secondary: 68N18

Rights: Copyright © 2006 University of Notre Dame


Vol.47 • No. 3 • 2006
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