Absolute convergence and Taylor expansion in web based models of Linear Logic

TL;DR

This paper develops a unified framework for web-based models of linear logic, extending Taylor expansion to include models with partial sums and non-positive coefficients, covering a broad class of spaces.

Contribution

It introduces a generic construction of web-based models with partial sums and generalizes Taylor expansion to models with non-positive coefficients, unifying various models.

Findings

All considered web models admit Taylor expansion.

The framework applies to coherence, probabilistic coherence, finiteness, and K{"o}the spaces.

Extension to models with non-positive coefficients.

Abstract

The differential $\lambda$-calculus studies how the quantitative aspects of programs correspond to differentiation and to Taylor expansion inside models of linear logic. Recent work has generalized the axioms of Taylor expansion so they apply to many models that only feature partial sums. However, that work does not cover the classic web based models of K{\"o}the spaces and finiteness spaces . First, we provide a generic construction of web based models with partial sums. It captures models, ranging from coherence spaces to probabilistic coherence spaces, finiteness spaces and K{\"o}the spaces. Second, we generalize the theory of Taylor expansion to models in which coefficients can be non-positive. We then use our generic web model construction to provide a unified proof that all the aforementioned web based models feature such Taylor expansion.