Linear Realisability and Implicative Algebras

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.

💡 Research Summary

The paper establishes, for the first time, a formal bridge between Girard’s linear realizability models and the classical realizability constructions of Kleene and Krivine. The authors build on Miquel’s framework of implicative algebras, which unifies intuitionistic and classical realizability by means of a complete lattice equipped with an implication operation and a distinguished “separator” that captures deductive closure. Recognizing that linear logic forbids structural rules such as weakening and contraction, they introduce a relaxed variant called a linear implicative algebra. In this setting the separator is restricted to a set of linear λ‑terms—terms in which each abstraction binds exactly one variable and each free variable occurs at most once—thereby mirroring the resource‑sensitive nature of linear logic.

The paper first revisits the standard definition of implicative algebras and shows how an applicative structure (a lattice with an application operation) is equivalent to an implicative structure via the adjunction between application and implication. This equivalence allows the authors to translate the usual typing judgments of the λ‑calculus into a semantic type system based on the lattice order. They then define linear realizability situations as triples (P, Ex, J) consisting of a set of programs, an associative composition Ex, and a measurement J satisfying a 2‑cocycle (trefoil) condition. By extending programs to “projects” (pairs of a real number and a program) they obtain a well‑behaved orthogonality relation and define types as bi‑orthogonal closures. The logical connectives of multiplicative linear logic—linear implication (⊸) and tensor (⊗)—are interpreted as specific constructions on these types, and the authors demonstrate that the usual linear‑logic equations hold in this setting.

Crucially, the authors show that every linear realizability model (including geometry of interaction, ludics, and interaction graphs) yields a linear implicative algebra. The separator of this algebra consists precisely of the linear λ‑terms that can be typed using the linear typing rules derived from the model’s orthogonality. Conversely, given a linear implicative algebra, one can reconstruct a linear realizability situation by interpreting the lattice elements as types and the separator as the set of admissible proofs. This bidirectional correspondence reveals that linear realizability models sit naturally between the intuitionistic separator (capturing Kleene‑style realizability) and the classical separator (capturing Krivine’s classical realizability), providing a “linear decomposition” of the traditional implicative algebra.

The paper concludes by discussing the implications of this decomposition for the construction of triposes. Since every implicative algebra gives rise to a tripos, the linear version suggests the possibility of a linear tripos—a categorical structure that would model linear logic in a way analogous to the well‑known realizability triposes for intuitionistic and classical logic. The authors outline several avenues for future work, including extending the framework to exponentials, exploring new classical realizability models derived from linear ones, and investigating set‑theoretic applications such as novel models of ZF set theory. Overall, the work unifies three major strands—Kleene’s intuitionistic realizability, Krivine’s classical realizability, and Girard’s linear realizability—under a single algebraic and categorical umbrella, opening the door to richer interactions between logic, computation, and category theory.