A general form of relative recursion

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.