An Analytic Propositional Proof System on Graphs

TL;DR

This paper introduces a novel proof system operating on arbitrary graphs, extending propositional logic beyond formulas and cographs, using modular decomposition and deep inference techniques.

Contribution

It develops a proof system for graphs that generalizes propositional logic, dropping cograph restrictions and employing modular decomposition and deep inference methods.

Findings

Admissibility of cut in the graph-based proof system

Generalization of the splitting property

Conservative extension of multiplicative linear logic with mix

Abstract

In this paper we present a proof system that operates on graphs instead of formulas. Starting from the well-known relationship between formulas and cographs, we drop the cograph-conditions and look at arbitrary undirected) graphs. This means that we lose the tree structure of the formulas corresponding to the cographs, and we can no longer use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalisation of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic with mix, and we argue that our graphs form a notion of generalised connective.