Graphs of Plural Cuts

TL;DR

This paper explores the structure of graphs generated by plural (multiple-conclusion) cuts in sequent systems, providing both inductive and combinatorial definitions, and analyzing their properties like planarity.

Contribution

It introduces a non-inductive, graph-theoretical definition of plural cut graphs, linking them to polycategories and characterizing their planarity.

Findings

Graphs of plural cuts are related to polycategories.

A combinatorial characterization of planarity is provided.

The paper connects sequent systems without permutation to specific graph structures.

Abstract

Plural (or multiple-conclusion) cuts are inferences made by applying a structural rule introduced by Gentzen for his sequent formulation of classical logic. As singular (single-conclusion) cuts yield trees, which underlie ordinary natural deduction derivations, so plural cuts yield graphs of a more complicated kind, related to trees, which this paper defines. Besides the inductive definition of these oriented graphs, which is based on sequent systems, a non-inductive, graph-theoretical, combinatorial, definition is given, and to reach that other definition is the main goal of the paper. As trees underlie multicategories, so the graphs of plural cuts underlie polycategories. The graphs of plural cuts are interesting in particular when the plural cuts are appropriate for sequent systems without the structural rule of permutation, and the main body of the paper deals with that matter. It gives a combinatorial characterization of the planarity of the graphs involved.