An algebraic extension of intuitionistic linear logic: the 𝓛!𝒮-calculus and its categorical model

Abstract We introduce the ${{\mathcal L}_!^{\mathcal S}}$-calculus, a linear lambda-calculus extended with scalar multiplication and term addition, that acts as a proof language for intuitionistic linear logic. These algebraic operations enable the direct expression of linearity at the syntactic level, a property not typically available in standard proof-term calculi. Building upon previous work, we develop the ${{\mathcal L}_{!}^{\mathcal S}}$-calculus as an extension of the ${\mathcal L}^{\mathcal S}$-calculus with the ! modality. We prove key meta-theoretical properties—subject reduction, confluence, strong normalization and an introduction property—as well as preserve the expressiveness of the original ${\mathcal L}^{\mathcal S}$-calculus, including the encoding of vectors and matrices, and the correspondence between proof-terms and linear functions. A denotational semantics is provided in the framework of linear categories with biproducts, ensuring a sound and adequate interpretation of the calculus. This work is part of a broader programme aiming to build a measurement-free quantum programming language grounded in linear logic.