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Ordered combinatory algebras and realizability†

Published online by Cambridge University Press:  10 September 2015

WALTER FERRER SANTOS ,
JONAS FREY ,
MAURICIO GUILLERMO ,
OCTAVIO MALHERBE  and
ALEXANDRE MIQUEL
WALTER FERRER SANTOS
Affiliation:
Centro de Matemática. Facultad de Ciencias. Universidad de la República, Montevideo, Uruguay
JONAS FREY
Affiliation:
Department of Computer Science. University of Copenhagen. Universitetsparken 1. København Ø
MAURICIO GUILLERMO
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia. Facultad de Ingeniería. Universidad de la República, Montevideo, Uruguay
OCTAVIO MALHERBE
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia. Facultad de Ingeniería. Universidad de la República, Montevideo, Uruguay
ALEXANDRE MIQUEL
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia. Facultad de Ingeniería. Universidad de la República, Montevideo, Uruguay
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Abstract

We propose the new concept of Krivine ordered combinatory algebra ($\mathcal{^KOCA}$) as foundation for the categorical study of Krivine's classical realizability, as initiated by Streicher (2013).

We show that $\mathcal{^KOCA}$'s are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. The difference between the two representations is that the elements of a $\mathcal{^KOCA}$ play both the role of truth values and realizers, whereas truth values are sets of realizers in $\mathcal{AKS}$s.

To conclude, we give a direct presentation of the realizability interpretation of a higher order language in a $\mathcal{^KOCA}$, which showcases the dual role that is played by the elements of the $\mathcal{^KOCA}$.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 3 , March 2017 , pp. 428 - 458
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

‡ This author would like to thank M. Guillermo and the uruguayan government agency ANII, for making possible his visit to Uruguay that resulted in a productive collaboration.
†

This work was partially supported by ANII (Ur), CSIC (Ur), IFUM (Fr/Ur), ANR (Fr).

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