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Categorical approach to horizontal structuring and refinement of high-level replacement systems. (English)
Appl. Categ. Struct. 7, No.4, 371-403 (1999).
Summary: Based on the well-known theory of high-level replacement systems ‒ a categorical formulation of graph grammars ‒ we present new results concerning refinement of high-level replacement systems. Motivated by Petri nets, where refinement is often given by morphisms, we give a categorical notion of refinement. This concept is called $Q$-transformations and is established within the framework of high-level replacement systems. The main idea is to supply rules with an additional morphism, which belongs to a specific class ${\cal Q}$ of morphisms. This leads to the new notions of ${\cal Q}$-rules and ${\cal Q}$-transformations. Moreover, several concepts and results of high-level replacement systems are extended to ${\cal Q}$-transformations. These are sequential and parallel transformations, union, and fusion, based on different colimit constructions. The main results concern the compatibility of these constructions with ${\cal Q}$-transformations that is the corresponding theorems for usual transformations are extended to ${\cal Q}$-transformations. Finally, we demonstrate the application of these techniques for the special case of Petri nets to a case study concerning the requirements engineering of a medical information system.