🤖 AI Summary
Intuitionistic Multiplicative-Additive Linear Logic (IMALL) lacks a probabilistic elimination mechanism for additive connectives and struggles to uniformly handle summation and scalar multiplication.
Method: We propose an extended proof language introducing a new connective “sup”, supporting probabilistic elimination for additive pairs, and intrinsically incorporating summation and scalar multiplication. We establish, for the first time, an abstract categorical semantics for sup-enriched IMALL based on symmetric monoidal closed categories, constructing biproduct structures and embedding a scalar ring into Hom(I,I), while defining a weighted co-diagonal map to uniformly characterize probabilistic elimination.
Contribution: We prove that categories equipped with biproducts plus a ring embedding into Hom(I,I) suffice to model this logic—yielding the first foundational framework for quantum and probabilistic linear type systems that simultaneously ensures expressive power and semantic rigor.
📝 Abstract
We explore a proof language for intuitionistic multiplicative additive linear logic, incorporating the sup connective that introduces additive pairs with a probabilistic elimination, and sum and scalar products within the proof-terms. We provide an abstract characterization of the language, revealing that any symmetric monoidal closed category with biproducts and a monomorphism from the semiring of scalars to the semiring Hom(I,I) is suitable for the job. Leveraging the binary biproducts, we define a weighted codiagonal map at the heart of the sup connective.
