[Paper Review] A Report on Realizability

Walter Ferrer Santos, Mauricio Guillermo+1 more|arXiv (Cornell University)|Sep 3, 2013

Semantic Web and Ontologies4 references3 citations

TL;DR

This paper introduces Krivine's Ordered Combinatory Algebras ($\mathcal{^{K}OCA}$) as a unified foundational framework that unifies computational and categorical semantics in realizability. By extending Abstract Krivine Structures with an adjunctor and completeness conditions, the authors show how $\mathcal{^{K}OCA}$ algebras directly induce Triplos and support realizability for higher-order languages, including higher-order Peano arithmetic, thereby enabling both proof-theoretic and categorical semantics within a single algebraic structure.

ABSTRACT

Besides recalling the basic definitions of Realizability Lattices, Abstract Krivine Structures, Ordered Combinatory Algebras and Tripos and reviewing its relationships, we propose a new foundational framework for realizability. Motivated by Streicher's paper "Krivine's Classical Realizability from a Categorical Perspective" [9], we define the concept of Krivine's Ordered Combinatory Algebras (kOKA) as a common platform that is strong enough to do both: categorical and computational semantics. The OCAs produced by Streicher from AKSs in [9] are particular cases of kOKAs.

Motivation & Objective

• To unify computational and categorical semantics in realizability theory by introducing a common algebraic foundation.
• To generalize Streicher's construction of Ordered Combinatory Algebras (OCAs) from Abstract Krivine Structures (AKS) into a broader class of algebras.
• To show that $\mathcal{^{K}OCA}$ algebras—equipped with an adjunctor and completeness—can directly induce Triplos without retracing to AKS.
• To establish realizability for higher-order languages, including higher-order Peano arithmetic, within the $\mathcal{^{K}OCA}$ framework.
• To demonstrate that key axioms of Peano Arithmetic, except induction, are realizable in any $\mathcal{^{K}OCA}$.

Proposed method

• Define Krivine's Ordered Combinatory Algebras ($\mathcal{^{K}OCA}$) as a generalization of OCAs with an adjunctor ensuring logical implication is internalized.
• Introduce the adjunctor $\operatorname{e}$ such that $a \circ b \leq c$ implies $\operatorname{e} \circ a \leq b \rightarrow c$, enabling internalization of implication.
• Construct $\mathcal{^{K}OCA}$ from Abstract Krivine Structures (AKS), showing that Streicher's OCAs are special cases.
• Impose a completeness condition with respect to arbitrary infima on $\mathcal{^{K}OCA}$ to directly induce a Tripos from the algebra.
• Define a realizability interpretation for higher-order languages $\mathcal{L}^\omega$, including $\mathcal{L}^\omega$-terms and formulas.
• Verify that equational axioms of Peano Arithmetic are realizable in $\mathcal{^{K}OCA}$ using specific realizers like $\lambda^*xx$ and $\lambda^*x\,x\operatorname{s}$.

Experimental results

Research questions

• RQ1 Can a single algebraic structure unify both computational and categorical semantics in realizability theory?
• RQ2 How can the adjunctor in an OCA be used to internalize implication and support logical reasoning?
• RQ3 Can a Tripos be directly induced from an $\mathcal{^{K}OCA}$ without relying on an intermediate AKS?
• RQ4 Are the equational axioms of higher-order Peano Arithmetic realizable in $\mathcal{^{K}OCA}$?
• RQ5 Can realizability for full higher-order logic be defined within the $\mathcal{^{K}OCA}$ framework?

Key findings

• The class of $\mathcal{^{K}OCA}$ algebras provides a common platform that supports both computational and categorical semantics in realizability.
• Every Abstract Krivine Structure (AKS) gives rise to a $\mathcal{^{K}OCA}$ with an adjunctor, generalizing Streicher's construction.
• With completeness under arbitrary infima, an $\mathcal{^{K}OCA}$ directly induces a Tripos, bypassing the need to return to the AKS.
• The identity term $\lambda^*xx$ realizes all equational axioms of Peano Arithmetic in any $\mathcal{^{K}OCA}$.
• The term $\lambda^*x\,x\operatorname{s}$ realizes the axiom $\forall x^I(\operatorname{succ}^I x \neq 0^I)$ in $\mathcal{^{K}OCA}$.
• Realizability for higher-order Peano Arithmetic, including the definition of $\mathds{N}(z^I)$ via Leibniz equality, is fully supported in $\mathcal{^{K}OCA}$.