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$!$带来了非线性证明与线性证明之间的区分，其中“线性”意味着每个论证恰好使用一次。微分线性逻辑（$\mathsf{DiLL}$）是线性逻辑的一种扩展，它包含了为$!$增加的新规则，这些规则编码了微分过程以及将证明线性化的能力。另一方面，分级线性逻辑（$\mathsf{GLL}$）是线性逻辑的一种变体，其中$!$现在由一个半环$R$进行索引。这种$R$-分级允许度为$r \in R$的非线性证明，而线性证明的度为$1 \in R$。近年来，人们开始关注将这两种$\mathsf{LL}$的变体相结合，并发展分级微分线性逻辑（$\mathsf{GDiLL}$）。在本文中，我们提出了$\mathsf{GDiLL}$的相继演算，并引入了其范畴语义，即我们称之为分级微分范畴，该语义同时使用了余余运算和导子变换。我们证明了对称幂总能产生分级微分范畴，并提供了分级微分范畴的其他例子。我们还讨论了（幺半）余代数模态、加性双代数模态以及Seely同构的分级版本，以及它们在$\mathsf{GDiLL}$的相继演算中的实现。