An Outline of Rule-Based Re nement for Petri Nets

Extended Abstract The concept of reenement is a well-known technique within software engineering in general and for stepwise development of Petri nets in particular. In fact, several diierent concepts for the reenement of nets have been proposed in literature, above all the reenement of places, transitions or even subnets by some other subnet. Based on the idea of formal grammars, we propose rule-based reenement to present rules denoting the replacement of a subnet by another one, without changing the remaining part of the whole net. This has the advantage of a simple local presentation of the reenement, even if the whole system is large and complex. We consider to have a rule p with a left-hand side net L that is replaced by a right-hand side net R. This rule can be applied to some net N, yielding the new net M. This application of a rule 1 p, called transformation 1 , is denoted by N p =) M. We study rule-based reenement and horizontal structuring for Petri nets in general. This means, that in Pad96] we have introduced a notion of abstract Petri nets, that comprises several diierent Petri net classes, basic as well as high-level. A general approach to rule-based reenement has already been introduced ((Pad96]) by the theory of high-level replacement systems in the sense of EHKP91b, EHKP91a]. This theory generalizes the theory of graph grammars to high-level structures deened as objects in a suitable category. We discuss the relation of rule-based reenement to other notions of reenement. Moreover, we review the concepts and results for independence and parallelism of transformations from EHKP91b, EHKP91a]. We introduce horizontal structuring techniques, union and fusion, that are expressed in the frame of high-level replacement systems using colimit constructions. Furthermore, we sketch the compatibility of these constructions with transformations. has strongly innuenced this work. 1 In the standard terminologyof graph grammars rules and transformations are called productionsand derivations, respectively. In the context of nets, however, we prefer to speak about rules and transformations.