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，其中线性意味着每个论元恰好使用一次。微分线性逻辑（$\mathsf{DiLL}$）是线性逻辑的扩展，包含针对$!$的附加规则，用以编码微分及证明线性化的能力。另一方面，分级线性逻辑（$\mathsf{GLL}$）是线性逻辑的一种变体，其中$!$现在由半环$R$索引。这种$R$-分级允许度$r \in R$的非线性证明，而线性证明对应度$1 \in R$。近年来，将这两种$\mathsf{LL}$变体结合并发展出分级微分线性逻辑（$\mathsf{GDiLL}$）的研究备受关注。本文给出了$\mathsf{GDiLL}$的相继式演算，并引入其范畴语义——我们称为分级微分范畴——同时利用余消去变换与推导变换。我们证明了对称幂总能生成分级微分范畴，并给出了分级微分范畴的其他实例。此外，本文还讨论了（幺半）余代数模态、加法双代数模态以及Seely同构的分级版本，及其在$\mathsf{GDiLL}$相继式演算中的实现。