Source: Notre Dame J. Formal Logic Volume 47, Number 3
(2006), 311-318.
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.
References
[1] Bethke, I., Notes on Partial Combinatory Algebras, Ph.D. thesis, Universiteit van Amsterdam, 1988.
[2] Hofstra, P., and J. van Oosten, "Ordered partial combinatory algebras", Mathematical Proceedings of the Cambridge Philosophical Society, vol. 134 (2003), pp. 445--63.
[3] Hofstra, P., and J. van Oosten, "Erratum to `Ordered partial combinatory algebras'", (2003). http://www.math.uu.nl/people/jvoosten/realizability/erratum.ps.
[4] Hyland, J. M. E., "The effective topos", pp. 165--216 in The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), edited by A. S. Troelstra and D. van Dalen, vol. 110 of Studies in Logic the Foundations of Mathematics, North-Holland, Amsterdam, 1982.
Mathematical Reviews (MathSciNet):
MR717245
[5] Longley, J., Realizability Toposes and Language Semantics, Ph.D. thesis, Edinburgh University, 1995.
[6] van Oosten, J., "A combinatory algebra for sequential functionals of finite type", pp. 389--405 in Models and Computability (Leeds, 1997), edited by S. B. Cooper and J. K. Truss, vol. 259 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999.
[7] Phoa, W., "Relative computability in the effective topos", Mathematical Proceedings of the Cambridge Philosophical Society, vol. 106 (1989), pp. 419--22.
