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 $\mathsf{GDiLL}$.

翻译：在线性逻辑（$\mathsf{LL}$）中，指数模态$!$区分了非线性证明与线性证明，其中“线性”意味着参数恰好使用一次。微分线性逻辑（$\mathsf{DiLL}$）是线性逻辑的扩展，它包含了针对$!$的额外规则，这些规则编码了微分以及线性化证明的能力。另一方面，分级线性逻辑（$\mathsf{GLL}$）是线性逻辑的一种变体，其中$!$现在以半环$R$作为索引。这种$R$-分级允许程度$r \in R$的非线性证明，而线性证明则对应程度$1 \in R$。近期，人们有兴趣将这两种$\mathsf{LL}$的变体相结合，发展出分级微分线性逻辑（$\mathsf{GDiLL}$）。本文提出了$\mathsf{GDiLL}$的矢列演算，并介绍了其范畴语义——我们称之为分级微分范畴，该语义同时使用了余消去变换和导数变换。我们证明了对称幂总是给出分级微分范畴，并提供了其他分级微分范畴的实例。此外，我们还讨论了（幺半群）余代数模态、加法双代数模态以及Seely同构的分级版本，以及它们在$\mathsf{GDiLL}$矢列演算中的实现。