Graded Differential Categories and Graded Differential Linear Logic

TL;DR

This work introduces Graded Differential Linear Logic (GDiLL) by unifying Differential Linear Logic with Graded Linear Logic, defining $R$-graded coalgebra modalities and $R$-graded differential modalities, and establishing a sequent calculus and semantics via graded differential categories. It proves the equivalence of deriving transformations and coderelictions in the graded setting and develops graded Seely isomorphisms and graded additive bialgebra modalities to support a cohesive structure for differentiation with grading. Concrete models are provided, notably symmetric powers and the REL category, showing how graded structures yield finite-dimensional, tractable differential semantics and offering a path toward applications in categorical quantum mechanics. The paper thereby lays foundational theory and exhibits rich examples that illuminate the interaction between grading and differentiation in a categorical context, with numerous avenues for future exploration (e.g., Taylor expansions, order structures, and additional models).

Abstract

In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.

Theorems & Definitions (17)

• definition 1
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• definition 3
• lemma 1
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• definition 4
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• definition 6
• theorem 1
• proof