Yeo's Theorem for Locally Colored Graphs: the Path to Sequentialization in Linear Logic

2026-03-03

AI Summary

This study addresses the problem of efficiently extracting sequent calculus derivations from linear logic proof nets without altering their underlying graph structure. To this end, we propose a local coloring graph model based on half-edge coloring, which generalizes Yeo’s theorem to uniformly identify splitting vertices. Our approach integrates the Danos–Regnier correctness criterion, a cusp minimization lemma, and path analysis techniques, enabling modular reconstruction of sequent derivations directly from the original proof net. The method applies to linear logic systems featuring the mix rule and lacking multiplicative units. Beyond unifying and simplifying several existing graph-theoretic results, our work establishes that, in the absence of cusp cycles, a splitting vertex always exists, thereby guaranteeing an efficient sequentialization procedure.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We revisit sequentialization proofs associated with the Danos-Regnier correctness criterion in the theory of proof nets of linear logic. Our approach relies on a generalization of Yeo's theorem for graphs, based on colorings of half-edges. This happens to be the appropriate level of abstraction to extract sequentiality information from a proof net without modifying its graph structure. We thus obtain different ways of recovering a sequent calculus derivation from a proof net inductively, by relying on a splitting vertex, which we can impose to be a par-vertex, or a terminal vertex, or a non-axiom vertex, etc., in a modular way. This approach applies in presence of the mix-rules as well as for proof nets of unit-free multiplicative-additive linear logic (through an appropriate further generalization of Yeo's theorem). The proof of our Yeo-style theorem relies on a key lemma that we call cusp minimization. Given a coloring of half-edges, a cusp in a path is a vertex whose adjacent half-edges in the path have the same color. And, given a cycle with at least one cusp and subject to suitable hypotheses, cusp minimization constructs a cycle with strictly less cusps. In the absence of cusp-free cycles, cusp minimization is then enough to ensure the existence of a splitting vertex, i.e. a vertex that is a cusp of any cycle it belongs to. Our theorem subsumes several graph-theoretical results, including some known to be equivalent to Yeo's theorem. The novelty is that they can be derived in a straightforward way, just by defining a dedicated coloring, again without any modification of the underlying graph structure (vertices and edges) -- similar results from the literature required more involved encodings.

Problem

• sequentialization
• proof nets
• linear logic
• Yeo's theorem
• graph coloring

Innovation

• proof nets
• sequentialization
• Yeo's theorem
• half-edge coloring
• cusp minimization