# Compositionality of Rewriting Rules with Conditions

## TL;DR

This paper extends the framework of compositional rewriting rules to include conditions, using category theory to analyze rule composition in DPO and SqPO rewriting systems with application conditions.

## Contribution

It introduces a category-theoretical approach to rule composition with conditions, demonstrating associativity and compatibility in complex rewriting systems.

## Key findings

- Established a formal framework for rule composition with conditions
- Proved associativity of rule composition in the extended setting
- Analyzed the interplay between conditions and categorical constructions

## Abstract

We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon $\mathcal{M}$-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.