[Paper Review] Realizability in OCAs and AKSs

Walter Ferrer Santos, Mauricio Guillermo+1 more|arXiv (Cornell University)|Dec 24, 2015

Atrial Fibrillation Management and Outcomes5 references3 citations

TL;DR

This paper introduces a new closure operator and associated maps in ordered combinatory algebras (OCAs) and abstract Krivine structures (AKSs), enabling a full adjunction between application and implication operations. By constructing a new OCA, denoted $\mathcal{A}_{\mathcal{K},\bullet}$, it achieves stronger adjunction properties than previous constructions, allowing for a simplified and more direct derivation of triposes from AKSs, thus providing a cleaner categorical realization of Krivine's classical realizability.

ABSTRACT

In the context of the $\mathcal{OCA}$ associated to an ${\mathcal{AKS}}$ we introduce a closure operator and two associated maps that replace the closure and the maps defined in \cite{kn:ocar}. We were motivated by the search of a full adjunction to the original implication map. We show that all the constructions from $\mathcal{OCA}$s to triposes developped in \cite{kn:ocar} can be also implemented in the new situation.

Motivation & Objective

• To resolve the lack of full adjunction between application and implication in earlier realizability constructions.
• To develop a new closure operator and associated maps that improve the categorical behavior of realizability structures.
• To construct a new OCA, $\mathcal{A}_{\mathcal{K},\bullet}$, from an AKS that supports full adjunction via an 'adjunctor' term.
• To show that this new OCA yields a tripos equivalent to those derived from earlier methods, simplifying the categorical realization of classical realizability.
• To provide a framework where the standard constructions from OCAs to triposes can be replicated with improved structural properties.

Proposed method

• Introduce a new closure operator $(-)\,\widehat{}$ and a bounded product $\sharp$ on an OCA, replacing the original closure and maps from [5].
• Define realizability lattices using polarity and closure operators, including the double perpendicular and the new $(-)\,\widehat{}$ operator.
• Introduce three pairs of conductor and push operations (application/implication-like) to better capture adjunction behavior.
• Define a new OCA $\mathcal{A}_{\mathcal{K},\bullet}$ using a distinguished 'adjunctor' term to recover full adjunction between application and implication.
• Construct an indexed preorder $\mathbf{P}_{\bullet}(\mathcal{K})$ from an AKS $\mathcal{K}$, and show it is isomorphic to $\mathbf{P}(\mathcal{A}_{\mathcal{K},\bullet})$.
• Prove equivalence between triposes derived from $\mathcal{A}_{\mathcal{K},\bullet}$ and those from earlier constructions, using commutativity of the relevant diagram up to equivalence.

Experimental results

Research questions

• RQ1 Can a full adjunction between application and implication be achieved in realizability structures without relying on external assumptions?
• RQ2 How can the closure operators and maps in OCAs be redefined to improve adjunction behavior?
• RQ3 What structural changes in an AKS or OCA are necessary to support a direct construction of a tripos with full adjunction?
• RQ4 Is the new OCA $\mathcal{A}_{\mathcal{K},\bullet}$ equivalent to existing constructions in terms of the resulting tripos?
• RQ5 Can the standard path from OCAs to triposes be preserved while improving the categorical properties of the construction?

Key findings

• The new closure operator $(-)\,\widehat{}$ and associated maps provide a better-behaved alternative to the double perpendicular construction in [5] and [13].
• The introduction of the 'adjunctor' term in $\mathcal{A}_{\mathcal{K},\bullet}$ enables full adjunction between application and implication, resolving a key limitation in earlier constructions.
• The tripos $\mathbf{P}_{\bullet}(\mathcal{K})$ derived from an AKS $\mathcal{K}$ is isomorphic to the tripos $\mathbf{P}(\mathcal{A}_{\mathcal{K},\bullet})$ derived from the new OCA, establishing equivalence.
• The construction of triposes from $\mathcal{A}_{\mathcal{K},\bullet}$ follows the same path as in [5], but now with stronger categorical properties due to the improved adjunction.
• The canonical inclusion $\mathbf{P}_{\bullet}(\mathcal{K}) \hookrightarrow \mathbf{P}(\mathcal{K})$ is an equivalence of indexed preorders, confirming consistency with standard realizability semantics.
• The diagram of constructions commutes up to equivalence, guaranteeing that $\mathbf{P}(\mathcal{A})$ and $\mathbf{P}(\mathcal{A}_{\mathcal{K}_{\mathcal{A}\bullet}\bullet})$ are equivalent triposes.