Unifying Graded Linear Logic and Differential Operators
This paper addresses the challenge of unifying graded linear logic (GLL) with differential operator semantics to simultaneously capture resource sensitivity and program differentiation behavior. Methodologically, we introduce **Graded Differential Linear Logic (GDL-LC)**—the first logical system that employs the differential operator monad as a grading index for exponential modalities, thereby intrinsically aligning resource accounting with proof linearization. Our formal system is built upon constant-coefficient linear partial differential operators, distribution theory, and monadic algebra; we establish its consistency and construct the first denotational model valued in generalized functions. The results show that GDL-LC is equivalent to the graded variant of finite differential linear logic, enabling fine-grained program complexity analysis while providing the first logical foundation and semantic framework for resource-aware differential computation.