Analytical Differential Calculus with Integration

TL;DR

This paper introduces a novel differential calculus with integration, bridging programming and mathematics, and explores its applications in incremental computation, automatic differentiation, and approximation.

Contribution

It presents the first differential calculus with integration from a programming perspective, maintaining mathematical rigor and extending the calculus's applicability.

Findings

Constructed reduction rules that align with mathematical principles

Preserved key mathematical theorems within the calculus

Demonstrated applications in automatic differentiation and incremental computation

Abstract

Differential lambda-calculus was first introduced by Thomas Ehrhard and Laurent Regnier in 2003. Despite more than 15 years of history, little work has been done on a differential calculus with integration. In this paper, we shall propose a differential calculus with integration from programming point of view. We show its good correspondence with mathematics, which is manifested by how we construct these reduction rules and how we preserve important mathematical theorems in our calculus. Moreover, we highlight applications of the calculus in incremental computation, automatic differentiation, and computation approximation.