[Paper Review] The Sup Connective in IMALL: A Categorical Semantics

Alejandro Díaz-Caro, Octavio Malherbe|arXiv (Cornell University)|May 4, 2022

Logic, Reasoning, and Knowledge4 citations

TL;DR

This paper establishes a categorical semantics for the $\mathcal{L}^{\mathcal{S}}$-calculus, a fragment of intuitionistic linear logic with syntactic linearity via sum and scalar multiplication. It shows that the calculus is sound and adequate when interpreted in the category of cancellative semimodules over a commutative semiring $\mathcal{S}$, generalizing prior work over fields to a broader algebraic foundation.

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 characterisation 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 which is at the core of the sup connective.

Motivation & Objective

• To extend the algebraic foundation of the $\mathcal{L}^{\mathcal{S}}$-calculus beyond fields to commutative semirings.
• To provide a categorical semantics for the $\mathcal{L}^{\mathcal{S}}$-calculus in the category of cancellative $\mathcal{S}$-semimodules.
• To prove that this semantics is both sound and adequate for the calculus.
• To clarify the minimal algebraic structure required to model syntactic linearity in proof terms.
• To lay the groundwork for interpreting quantum-inspired linear logic calculi in more general algebraic settings.

Proposed method

• The calculus is extended to use a commutative semiring $\mathcal{S}$ instead of a field, preserving interstitial rules for sum and scalar multiplication.
• The semantics is defined in the category $\mathbf{SM}_{\mathcal{S}}$ of cancellative $\mathcal{S}$-semimodules, which supports monoidal closure.
• The interpretation maps propositions to $\mathcal{S}$-semimodules and proofs to linear maps between them.
• The soundness and adequacy of the semantics are proven via structural induction on proof terms and by verifying commutativity of key diagrams.
• The cancellative property of $\mathcal{S}$ is essential for the adjunction between tensor product and hom-functor, ensuring categorical closure.
• The construction uses the standard categorical isomorphisms $\lambda$, $\rho$, and $\delta$ to model structural rules in the semantics.

Experimental results

Research questions

• RQ1 What is the minimal algebraic structure required to model the $\mathcal{L}^{\mathcal{S}}$-calculus categorically?
• RQ2 Can the $\mathcal{L}^{\mathcal{S}}$-calculus be given a sound and adequate categorical semantics over a commutative semiring?
• RQ3 Why is the cancellative property of the semiring $\mathcal{S}$ essential for the semantics to be well-behaved?
• RQ4 How does the semantics handle the interstitial rules for sum and scalar multiplication in the proof terms?
• RQ5 Can the semantics support the linear behavior of proof terms, such as $t(u+v) \equiv tu + tv$ and $t(s\bullet u) \equiv s\bullet tu$?

Key findings

• The $\mathcal{L}^{\mathcal{S}}$-calculus is sound and adequate when interpreted in the category of cancellative $\mathcal{S}$-semimodules for any commutative semiring $\mathcal{S}$.
• The cancellative property of $\mathcal{S}$ is necessary to ensure the existence of the adjunction between the tensor product and the internal hom, which is crucial for the categorical closure.
• The semantics correctly models syntactic linearity: $t(u \mathbin{\text{\scalebox{0.7}{\faPlus}}} v) \equiv tu \mathbin{\text{\scalebox{0.7}{\faPlus}}} tv$ and $t(s \bullet u) \equiv s \bullet tu$ hold in the model.
• The interpretation of $\top^n \Rightarrow \top^m$ corresponds exactly to linear maps $\mathcal{S}^n \to \mathcal{S}^m$, with proof terms representing such maps faithfully.
• The semantics supports the full linearity of the calculus, including the behavior of conjunction and disjunction, via appropriate diagram commutativity in the category.
• The results generalize prior work over fields to semirings, showing that the field structure is not essential for the semantics, only the cancellative semiring structure.