On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation
Parallel independence between transformation steps is a basic and well-understood notion of the algebraic approaches to graph transformation, and typically guarantees that the two steps can be applied in any order obtaining the same resulting graph, up to isomorphism. The concept has been redefined for several algebraic approaches as variations of a classical “algebraic” condition, requiring that each matching morphism factorizes through the context graphs of the other transformation step. However, looking at some classical papers on the double-pushout approach, one finds that the original definition of parallel independence was formulated in set-theoretical terms, requiring that the intersection of the images of the two left-hand sides in the host graph is contained in the intersection of the two interface graphs. The relationship between this definition and the standard algebraic one is discussed in this position paper, both in the case of left-linear and non-left-linear rules.
The idea of spelling out the relationship between the standard algebraic and the pullback-based definitions of parallel independence maturated during stimulating discussions with Dominque Duval, Frédéric Prost, Rachid Echahed and Leila Ribeiro, during the work on the AGREE approach to GT. Hans-Jörg Kreowski provided me some references to the early literature on parallel independence. During the workshop where this work was presented, Michael Löwe suggested several technical improvements, including a new version of the last part of the proof of Proposition 2 that does not need partial maps classifiers: this will be presented in a forthcoming report.
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