Single Pushout Rewriting in Comprehensive Systems

Abstract

The elegance of the single-pushout (SPO) approach to graph transformations arises from substituting total morphisms by partial ones in the underlying category. Thus, SPO's applicability depends on the durability of pushouts after this transition. There is a wide range of work on the question when pushouts exist in categories with partial morphisms starting with the pioneering work of Löwe and Kennaway and ending with an essential characterisation in terms of an exactness property (for the interplay between pullbacks and pushouts) and an adjointness condition (w.r.t. inverse image functions) by Hayman and Heindel. Triple graphs and graph diagrams are frameworks to synchronize two or more updatable data sources by means of internal mappings, which identify common sub-structures. Comprehensive systems generalise these frameworks, treating the network of data sources and their structural interrelations as a homogeneous comprehensive artifact, in which partial maps identify commonalities. Although this inherent partiality produces amplified complexity, Heindel's characterisation still yields cocompleteness of the category of comprehensive systems equipped with closed partial morphisms and thus enables computing by SPO graph transformation.