Ordered combinatory algebras and realizability

TL;DR

This paper explores various combinatory structures related to Krivine realizability, demonstrating their equivalence in modeling higher-order logic through categorical structures called triposes, and introduces Krivine ordered combinatory algebras as foundational for this framework.

Contribution

It establishes the equivalence of different combinatory structures in modeling higher-order logic and introduces Krivine ordered combinatory algebras as a categorical foundation for Krivine's realizability.

Findings

All considered structures produce the same class of triposes.

Krivine ordered combinatory algebras can define realizability in higher-order languages.

The structures are equivalent up to the associated indexed preorders.

Abstract

We consider different classes of combinatory structures related to Krivine realizability. We show, in the precise sense that they give rise to the same class of triposes, that they are equivalent for the purpose of modeling higher-order logic. We center our attentions in the role of a special kind of Ordered Combinatory Algebras-- that we call the "Krivine ordered combinatory algebras" ($\mathcal{KOCA}$s)-- that we propose as the foundational pillars for the categorical perspective of Krivine's classical realizability as presented by Streicher. Our procedure is the following: we show that each of the considered combinatory structures gives rise to an indexed preorder, and describe a way to transform the different structures into each other that preserves the associated indexed preorders up to equivalence. Since all structures give rise to the same indexed preorders, we only prove that they are triposes once: for the class of $\mathcal{KOCA}$s. We finish showing that in $\mathcal{KOCA}$s, one can define realizability in every higher-order language and in particular in higher-order arithmetic.