Morphisms between graph-transformation systems (GTSs) have been successfully used for the refinement, reuse, and composition of GTSs. All these uses share a fundamental problem: to be able to define a morphism, source and target GTSs need to be quite similar in their structure (in terms of both the type graphs and the set of rules and their respective structures). This limits the applicability of these approaches by excluding a wide range of mappings that would intuitively be accepted as meaningful, but that cannot be captured formally as a morphism. Some researchers have attempted to introduce some flexibility, but these attempts either focus only on the type graphs (e.g., Kleisli morphisms between type graphs) or only support specific forms of deviation (e.g., supporting sub-typing in type graphs through clan morphisms). In this work, we introduce the notion of GTS families, which provide a general mechanism for explicitly expressing the amount of acceptable adaptability of the involved GTSs so that the intended morphisms can be defined. On this basis, we demonstrate how GTS families that are extension preserving can be used to enable flexible GTS amalgamation.
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