Five Basic Concepts of Axiomatic Rewriting Theory

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 Lévy, 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. A good way to understand Axiomatic Rewriting Theory is to think of it as a 2-dimensional refinement of Abstract Rewriting Theory. Recall that an abstract rewriting system is defined as a set V of vertices (= terms) equipped with a binary relation → ⊆ V × V. This abstract formulation is convenient to formulate various notions of termination and of confluence, and to compare them, typically: strong normalisation vs. weak normalisation confluence vs. local confluence Unfortunately, the theory is not sufficiently informative to capture more sophisticated structures and properties of rewriting systems related to causal-ity and residuation, like redexes and residuals finite developments standardisation head rewriting paths These structures and properties are ubiquitous in rewriting theory. They appear in conflict-free rewriting systems like the λ-calculus as well as in rewriting systems with critical pairs, like action calculi and bigraphs designed by Milner [9] as universal calculus integrating the λ-calculus, Petri nets and process calculi, or the λσ-calculus introduced by Abadi, Cardelli, Curien and Lévy [1] to express in a single rewriting system the various evaluation strategies of an environment machine.