Affine logic with the integration operator

TL;DR

This paper extends affine continuous logic to include an integration operator, proving key theorems like the affine compactness theorem and establishing a completeness theorem, thereby advancing the theoretical framework of affine logic.

Contribution

It introduces affine integration logic, proves the affine compactness theorem, and establishes a completeness theorem, expanding the scope of affine continuous logic.

Findings

Affine compactness theorem proved via ultramean and Henkin's method

A completeness theorem for affine integration logic established

A variant of Keisler-Shelah isomorphism theorem holds in this setting

Abstract

Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant of the Keisler-Shelah isomorphism theorem holds in this setting. This helps us to characterize non-forking extensions in affine stable theories by means of the notion of elementary embedding in the expanded logic.