Rule Algebras for Adhesive Categories

TL;DR

This paper extends the theory of DPO graph rewriting by establishing associative rule composition in $ ext{M}$-adhesive categories and introduces rule algebras that enable analysis of stochastic rewriting systems.

Contribution

It generalizes the notion of rule composition to $ ext{M}$-adhesive categories and defines rule algebras with a canonical representation for stochastic rewriting analysis.

Findings

Associative sequential composition of rules in $ ext{M}$-adhesive categories.

Definition of unital rule algebras in categories with an $ ext{M}$-initial object.

Potential application to analyze evolution of statistical moments in stochastic rewriting.

Abstract

We demonstrate that the most well-known approach to rewriting graphical structures, the Double-Pushout (DPO) approach, possesses a notion of sequential compositions of rules along an overlap that is associative in a natural sense. Notably, our results hold in the general setting of $\mathcal{M}$-adhesive categories. This observation complements the classical Concurrency Theorem of DPO rewriting. We then proceed to define rule algebras in both settings, where the most general categories permissible are the finitary (or finitary restrictions of) $\mathcal{M}$-adhesive categories with $\mathcal{M}$-effective unions. If in addition a given such category possess an $\mathcal{M}$-initial object, the resulting rule algebra is unital (in addition to being associative). We demonstrate that in this setting a canonical representation of the rule algebras is obtainable, which opens the possibility of applying the concept to define and compute the evolution of statistical moments of observables in stochastic DPO rewriting systems.