Unique perfect matchings, edge-colored graphs and proof nets for linear logic with Mix

01/29/2019
by   Lê Thành Dũng Nguyên, et al.
0

This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retoré (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. Alternating cycles in perfect matchings serve as witnesses of non-uniqueness, and in this significantly expanded journal version, we discuss connections with other kinds of constrained cycles known to be equivalent: semicycles in directed graphs, trails avoiding forbidden transitions, and properly colored cycles in edge-colored graphs. While the second one provides an explanation and generalization of Retoré's "R&B-graphs", the latter leads us to prove the coNP-hardness of Pagani's visible acyclicity criterion for MELL proof nets. We also connect Lamarche's essential nets to R&B-graphs.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/23/2019

Complexity of correctness for pomset logic proof nets

We show that it is coNP-complete to decide whether a given proof structu...
research
01/21/2019

Constrained path-finding and structure from acyclicity

This note presents several results in graph theory inspired by the autho...
research
12/23/2019

Proof nets through the lens of graph theory: a compilation of remarks

This document is intended to eventually gather a few small remarks on th...
research
02/26/2019

A New Linear Time Correctness Condition for Multiplicative Linear Logic

In this paper, we give a new linear time correctness condition for proof...
research
07/03/2019

Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

Motivated by adjacency in perfect matching polytopes, we study the short...
research
08/17/2022

Simplicity in Eulerian Circuits: Uniqueness and Safety

An Eulerian circuit in a directed graph is one of the most fundamental G...

Please sign up or login with your details

Forgot password? Click here to reset