Multilevel Typed Graph Transformations

Multilevel modeling extends traditional modeling techniques with a potentially unlimited number of abstraction levels. Multilevel models can be formally represented by multilevel typed graphs whose manipulation and transformation are carried out by multilevel typed graph transformation rules. These rules are cospans of three graphs and two inclusion graph homomorphisms where the three graphs are multilevel typed over a common typing chain. In this paper, we show that typed graph transformations can be appropriately generalized to multilevel typed graph transformations improving preciseness, flexibility and reusability of transformation rules. We identify type compatibility conditions, for rules and their matches, formulated as equations and inequations, respectively, between composed partial typing morphisms. These conditions are crucial presuppositions for the application of a rule for a match---based on a pushout and a final pullback complement construction for the underlying graphs in the category Graph---to always provide a well-defined canonical result in the multilevel typed setting. Moreover, to formalize and analyze multilevel typing as well as to prove the necessary results, in a systematic way, we introduce the category Chain of typing chains and typing chain morphisms.