🤖 AI Summary
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.
📝 Abstract
Linear Logic refines Intuitionnistic Logic by taking into account the resources used during the proof and program computation. In the past decades, it has been extended to various frameworks. The most famous are indexed linear logics which can describe the resource management or the complexity analysis of a program. From an other perspective, Differential Linear Logic is an extension which allows the linearization of proofs. In this article, we merge these two directions by first defining a differential version of Graded linear logic: this is made by indexing exponential connectives with a monoid of differential operators. We prove that it is equivalent to a graded version of previously defined extension of finitary differential linear logic. We give a denotational model of our logic, based on distribution theory and linear partial differential operators with constant coefficients.