Rewriting Context-free Families of String Diagrams

TL;DR

This paper develops a formal, automatable framework for reasoning about infinite families of string diagrams using context-free graph grammars and double-pushout rewriting, enabling more efficient and error-resistant proofs.

Contribution

It introduces a novel mathematical framework that models equations between infinite string diagram families with context-free grammars and proves its soundness and decidability for software implementation.

Findings

Framework accurately models infinite diagram families.

Proves the framework's soundness with respect to string diagram semantics.

Establishes decidability properties for automated reasoning.

Abstract

String diagrams provide a convenient graphical framework which may be used for equational reasoning about morphisms of monoidal categories. However, unlike term rewriting, rewriting string diagrams results in shorter equational proofs, because the string diagrammatic representation allows us to formally establish equalities modulo any rewrite steps which follow from the monoidal structure. Manipulating string diagrams by hand is a time-consuming and error-prone process, especially for large string diagrams. This can be ameliorated by using software proof assistants, such as Quantomatic. However, reasoning about concrete string diagrams may be limiting and in some scenarios it is necessary to reason about entire (infinite) families of string diagrams. When doing so, we face the same problems as for manipulating concrete string diagrams, but in addition, we risk making further mistakes if we are not precise enough about the way we represent (infinite) families of string diagrams. The primary goal of this thesis is to design a mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation. We will be working with context-free families of string diagrams and we will represent them using context-free graph grammars. We will model equations between infinite families of diagrams using rewrite rules between context-free grammars. Our framework represents equational reasoning about concrete string diagrams and context-free families of string diagrams using double-pushout rewriting on graphs and context-free graph grammars respectively. We will prove that our representation is sound by showing that it respects the concrete semantics of string diagrammatic reasoning and we will show that our framework is appropriate for software implementation by proving important decidability properties.