Notre Dame Journal of Formal Logic

A General Form of Relative Recursion

Jaap van Oosten


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.

Article information

Notre Dame J. Formal Logic, Volume 47, Number 3 (2006), 311-318.

First available in Project Euclid: 17 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B40: Combinatory logic and lambda-calculus [See also 68N18]
Secondary: 68N18: Functional programming and lambda calculus [See also 03B40]

partial combinatory algebras realizability relative recursion


van Oosten, Jaap. A General Form of Relative Recursion. Notre Dame J. Formal Logic 47 (2006), no. 3, 311--318. doi:10.1305/ndjfl/1163775438.

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