!-Graphs with Trivial Overlap are Context-Free

TL;DR

This paper demonstrates that a significant subclass of !-graphs with trivial overlaps can be represented using context-free vertex replacement grammars, advancing the understanding of their formal properties.

Contribution

It introduces a formal encoding of !-graphs with trivial overlaps using context-free grammars, a novel approach in the study of their language properties.

Findings

Trivial-overlap !-graphs can be encoded with context-free grammars.

This encoding facilitates formal analysis of their properties.

The work advances understanding of the formal language aspects of !-graphs.

Abstract

String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be done automatically by double-pushout rewriting. !-graphs give us the means of expressing and proving properties about whole families of these graphs simultaneously. While !-graphs provide elegant proofs of surprisingly powerful theorems, little is known about the formal properties of the graph languages they define. This paper takes the first step in characterising these languages by showing that an important subclass of !-graphs--those whose repeated structures only overlap trivially--can be encoded using a (context-free) vertex replacement grammar.