Linear Realisability and Implicative Algebras

2026-02-06

AI Summary

This work establishes a formal connection between linear realizability models and classical realizability constructions such as those of Kleene and Krivine. Building on Miquel’s implicative algebra framework, we introduce a linear decomposition that positions linear logic as an intermediate layer between intuitionistic and classical logic, and we instantiate this decomposition semantically through a linear realizability model. This approach provides the first formal unification of linear realizability with traditional realizability theory, thereby extending the applicability of implicative algebras and offering a rigorous semantic bridge between distinct implementations of the BHK interpretation. Our results further illuminate the structural role of linear logic within realizability semantics.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Realizability, introduced by Kleene, can be understood as a concretization of the Brouwer-Heyting-Kolmogorov (BHK) interpretation of proofs, providing a framework to interpret mathematical statements and proofs in terms of their constructive or computational content. Over time, this concept has evolved through various extensions, such as Kreisel's modified realizability or Krivine's classical realizability. Parallel to these developments, Girard's work on linear logic introduced another perspective, often seen as another concrete realization of the BHK interpretation. The resulting constructions, encompassing models like geometry of interaction, ludics, and interaction graphs, were recently unified under the term linear realizability models to stress the intuitive connection with intuitionnistic and classical realizability. The present work establishes for the first time a formal link between linear realizability models and the realizability constructions of Kleene and Krivine. Our approach leverages Miquel's framework: just as linear logic can be viewed as a decomposition of intuitionistic and classical logic, we propose a linear decomposition of implicative algebras and show that linear realisability models provide concrete examples of such decompositions.

Problem

• linear realizability
• implicative algebras
• Kleene realizability
• Krivine realizability
• BHK interpretation

Innovation

• linear realizability
• implicative algebras
• BHK interpretation
• linear logic
• realizability models