# Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix

## TL;DR

This paper connects linear logic proof net correctness with perfect matching uniqueness in graph theory, providing new efficient algorithms and characterizations for MLL+Mix proof nets using graph algorithms.

## Contribution

It introduces a novel equivalence between proof net correctness and perfect matching uniqueness, leading to linear-time correctness criteria and new graph-theoretic insights.

## Key findings

- Linear-time correctness criterion for MLL+Mix proof nets
- Quasi-linear sequentialization algorithm
- Characterization of sub-polynomial complexity of correctness problem

## Abstract

This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retor\'e (2003). We show that the problem of correctness for MLL+Mix proof nets is equivalent to the problem of uniqueness of a perfect matching. By applying matching theory, we obtain new results for MLL+Mix proof nets: a linear-time correctness criterion, a quasi-linear sequentialization algorithm, and a characterization of the sub-polynomial complexity of the correctness problem. We also use graph algorithms to compute the dependency relation of Bagnol et al. (2015) and the kingdom ordering of Bellin (1997), and relate them to the notion of blossom which is central to combinatorial maximum matching algorithms. In this journal version, we have added an explanation of Retor\'e's "RB-graphs" in terms of a general construction on graphs with forbidden transitions. In fact, it is by analyzing RB-graphs that we arrived at this construction, and thus obtained a polynomial-time algorithm for finding trails avoiding forbidden transitions; the latter is among the material covered in another paper by the author focusing on graph theory (arXiv:1901.07028).