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

2026-03-26

AI Summary

This work addresses the limitations of existing Taylor expansion theory, which struggles to apply to classical web-based models of linear logic such as Köthe spaces and finiteness spaces, particularly when dealing with non-positive coefficients and partial summation structures. The paper introduces a general web-based semantic framework that accommodates partial summation and, for the first time, extends Taylor expansion theory to settings involving non-positive coefficients. This unified approach encompasses coherence spaces, probabilistic coherence spaces, finiteness spaces, and Köthe spaces. By integrating semantic tools from linear logic, differential λ-calculus, sequence space theory, and absolute convergence analysis of formal power series, the authors demonstrate that all major web-based models satisfy a generalized form of Taylor expansion, thereby broadening the mathematical foundations and applicability of differential program semantics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The differential $λ$-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{ö}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{ö}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.

Problem

• Taylor expansion
• Linear Logic
• web-based models
• absolute convergence
• Köthe spaces

Innovation

• Taylor expansion
• web-based models
• linear logic
• partial sums
• differential λ-calculus