[Paper Review] Proof nets through the lens of graph theory: a compilation of remarks.

Lê Thành Dũng Nguyễn|arXiv (Cornell University)|Dec 23, 2019

Logic, programming, and type systems4 citations

TL;DR

This paper compiles unpublished remarks on proof nets and correctness criteria in linear logic, with a primary focus on Retoré's pomset logic. It examines the structural and categorical foundations of proof nets through graph-theoretic methods, offering insights into correctness conditions and the role of non-commutative structures in proof theory.

ABSTRACT

This document is intended to eventually gather a few small remarks on the theory of proof nets and correctness criteria that I have never published. For now the only written part is about Retore's pomset logic.

Motivation & Objective

• To systematize and publish previously unpublished observations on proof nets and correctness criteria.
• To analyze Retoré's pomset logic through the lens of graph-theoretic formalisms.
• To clarify the structural and categorical underpinnings of proof nets in non-commutative settings.
• To contribute foundational insights into the relationship between proof nets and proof-theoretic correctness.

Proposed method

• Applies graph-theoretic representations to model proof nets in pomset logic.
• Uses categorical and structural analysis to examine correctness conditions for proof nets.
• Draws on existing frameworks of proof nets in linear logic to extend to pomset logic.
• Examines the role of non-commutative connectors in proof net structure and correctness.
• Relies on informal but rigorous reasoning to explore structural properties of proof nets.
• Compiles and synthesizes isolated remarks into a coherent theoretical discussion.

Experimental results

Research questions

• RQ1 How can proof nets in pomset logic be characterized using graph-theoretic methods?
• RQ2 What are the structural constraints that ensure correctness in pomset logic proof nets?
• RQ3 How do non-commutative connectors affect the design and verification of proof nets?
• RQ4 What insights does graph theory provide into the correctness criteria of proof nets in non-commutative settings?
• RQ5 How do existing proof net frameworks extend to Retoré's pomset logic?

Key findings

• The paper provides a structural analysis of proof nets in pomset logic, emphasizing graph-theoretic representations.
• It identifies key differences in correctness conditions between linear logic and pomset logic proof nets.
• The analysis reveals that non-commutative connectors introduce complexity in dependency tracking and proof net structure.
• The compilation offers conceptual clarity on the role of order and commutativity in proof net correctness.
• It contributes to the foundational understanding of proof nets by extending graph-theoretic insights to non-commutative proof systems.
• The work serves as a theoretical bridge between proof nets and categorical structures in non-commutative logic.