Double-Pullback Graph Transitions: A Rule-Based Framework with Incomplete Information

Reactive systems perform their tasks through interaction with their users or with other systems (as parts of a bigger system). An essential requirement for modeling such systems is the ability to express this kind of interaction. Classical rule-based approaches like Petri nets and graph transformation are not suited for this purpose because they assume to have complete control about the state and its transformation. Therefore, in this paper we propose a general framework which extends a given rule-based approach by a loose semantics where the rules of the system (e.g., graph productions or Petri net transitions) are considered as incomplete descriptions of the transformations to be performed: they still determine the changes to the matched substructure but for the remaining part (the context) unspecified changes are possible representing the application of (unknown) rules from the environment. The framework is applied to graph transformation systems in the doublepushout approach as well as place-transition Petri nets leading, respectively, to the concepts of graph transition and open step.