Five Basic Concepts of Axiomatic Rewriting Theory

TL;DR

This paper reviews five fundamental concepts of Axiomatic Rewriting Theory, a formal framework for understanding rewriting systems through axioms and diagrams, highlighting its development and current research directions.

Contribution

It provides a comprehensive overview of the axiomatic and diagrammatic approach to rewriting, integrating categorical semantics and higher-dimensional algebra.

Findings

Development of a 2-dimensional theory of causality and residuation

Integration of categorical semantics into rewriting theory

Identification of new research directions in the field

Abstract

In this invited talk, I will review five basic concepts of Axiomatic Rewriting Theory, an axiomatic and diagrammatic theory of rewriting started 25 years ago in a LICS paper with Georges Gonthier and Jean-Jacques L\'evy, and developed along the subsequent years into a full-fledged 2-dimensional theory of causality and residuation in rewriting. I will give a contemporary view on the theory, informed by my later work on categorical semantics and higher-dimensional algebra, and also indicate a number of current research directions in the field.