Formal Relationship between Petri Net and Graph Transformation Systems based on Functors between M-adhesive Categories

Various kinds of graph transformations and Petri net transformation systems are examples of M -adhesive transformation systems based on M -adhesive categories, generalizing weak adhesive HLR categories. For typed attributed graph transformation systems, the tool environment AGG allows the modeling, the simulation and the analysis of graph transformations. A corresponding tool for Petri net transformation systems, the RON-Environment, has recently been developed which implements and simulates Petri net transformations based on corresponding graph transformations using AGG. Up to now, the correspondence between Petri net and graph transformations is handled on an informal level. The purpose of this paper is to establish a formal relationship between the corresponding M -adhesive transformation systems, which allow the translation of Petri net transformations into graph transformations with equivalent behavior, and, vice versa, the creation of Petri net transformations from graph transformations. Since this is supposed to work for different kinds of Petri nets, we propose to define suitable functors, called M -functors, between different M -adhesive categories and to investigate properties allowing us the translation and creation of transformations of the corresponding M -adhesive transformation systems.