Adhesive category theory for graph rewriting in Rocq

2025-09-22

AI Summary

This work addresses the lack of formalization of adhesive categories—a foundational concept in graph rewriting theory—by developing the first structured, extensible formal library of adhesive categories in the Rocq proof assistant using Hierarchy Builder (HB). Methodologically, it introduces a two-layer hierarchical categorical architecture that systematically encapsulates morphisms and construction interfaces, supporting key categorical constructs such as pullbacks and equalizers, and instantiates adhesive structure across type categories, finite-type categories, simple graph categories, and presheaf categories. The main contributions are: (i) the first complete formalization of the axiomatic system of adhesive categories in Rocq; (ii) rigorous mechanized verification of the Church–Rosser and concurrency theorems for double-pushout rewriting; and (iii) empirical validation of HB’s scalability and maintainability in large-scale category-theoretic formalization. This work establishes a rigorous foundation for the formal semantics of graph rewriting and the verification of related tools.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We design a Rocq library about adhesive categories, using Hierarchy Builder (HB). It is built around two hierarchies. The first is for categories, with usual categories at the bottom and adhesive categories at the top, with weaker variants of adhesive categories in between. The second is for morphisms (notably isomorphisms, monomorphisms and regular monomorphisms). Each level of these hierarchies is equipped with several interfaces to define instances. We cover basic categorical concepts such as pullbacks and equalizers, as well as results specific to adhesive categories. Using this library, we formalize two central theorems of categorical graph rewriting theory: the Church-Rosser theorem and the concurrency theorem. We provide several instances, including the category of types, the category of finite types, the category of simple graphs and categories of presheaves. We detail the implementation choices we made and report on the usage of HB for this formalization work.

Problem

• Formalizing adhesive category theory for graph rewriting in Rocq
• Developing hierarchical library structures for categories and morphisms
• Proving Church-Rosser and concurrency theorems for graph rewriting

Innovation

• Developed adhesive category hierarchies using Hierarchy Builder
• Formalized graph rewriting theorems like Church-Rosser
• Implemented instances including categories of types and graphs