The algebras of graph rewriting

TL;DR

This paper introduces a new algebraic framework for analyzing graph rewriting systems using diagrammatic combinatorial Hopf algebras, revealing new types of graph rewriting systems and establishing foundational mathematical structures.

Contribution

It extends diagrammatic Hopf algebras to rule diagrams for graph rewriting, identifying new types of systems and proving structural theorems for the resulting algebras.

Findings

Existence of two new types of graph rewriting systems.

Development of rule algebras as non-commutative associative algebras.

A Poincaré-Birkhoff-Witt type theorem for rule diagram algebra.

Abstract

The concept of diagrammatic combinatorial Hopf algebras in the form introduced for describing the Heisenberg-Weyl algebra in~\cite{blasiak2010combinatorial} is extended to the case of so-called rule diagrams that present graph rewriting rules and their composites. The resulting rule diagram algebra may then be suitably restricted in four different ways to what we call the rule algebras, which are non-commutative, unital associative algebras that implement the algebra of compositions of graph rewriting rules. Notably, our framework reveals that there exist two more types of graph rewriting systems than previously known in the literature, and we present an analysis of the structure of the rule algebras as well as a form of Poincar\'{e}-Birkhoff-Witt theorem for the rule diagram algebra. Our work lays the foundation for a fundamentally new way of analyzing graph transformation systems, and embeds this very important concept from theoretical computer science firmly into the realm of mathematical combinatorics and statistical physics.