Structural Decomposition of Reactions of Graph-Like Objects

TL;DR

This paper introduces a method using colimits to decompose complex graph-like reactions into simpler parts within adhesive categories, enhancing understanding of compositionality in graph transformations.

Contribution

It formalizes the local decomposition problem for transformations and proves the soundness of colimit decomposition for double pushout transformations in adhesive categories.

Findings

Colimit decomposition is sound for arbitrary double pushout transformations.

A solution for a class of local decomposition problems is provided.

The work generalizes recent results on compositionality in graph transformation.

Abstract

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation.