Object-centric process mining investigates the intertwined behavior of multiple objects in business processes. From object-centric event logs, object-centric Petri nets (OCPN) can be discovered to replay the behavior of processes accessing different object types. Although they indicate how objects flow through the process and co-occur in events, OCPNs remain underspecified about the relationships of objects. Hence, they are not able to represent synchronization, i.e. executing objects only according to their intended relationships, and fail to identify violating executions. Existing formal modeling approaches, such as object-centric Petri nets with identifiers (OPID), represent object identities and relationships to synchronize them correctly. However, OPID discovery has not yet been studied. This paper uses explicit data models to bridge the gap between OCPNs and formal OPIDs. We identify the implicit assumptions of stable many-to-one relationships in object-centric event logs, which implies synchronization of related objects. To formally underpin this observation, we combine OCPNs with explicit stable many-to-one relationships in a rigorous mapping from OCPNs to OPIDs explicitly capturing the intended stable relationships and the synchronization of related objects. We prove that the original OCPNs and the resulting OPIDs coincide for those executions that satisfy the intended relationships. Moreover, we provide an implementation of the mapping from OCPN to OPID under stable relationships.

Traditional process mining considers only one single case notion and discovers and analyzes models based on this. However, a single case notion is often not a realistic assumption in practice. Multiple case notions might interact and influence each other in a process. Object-centric process mining introduces the techniques and concepts to handle multiple case notions. So far, such event logs have been standardized and novel process model discovery techniques were proposed. However, notions for evaluating the quality of a model are missing. These are necessary to enable future research on improving object-centric discovery and providing an objective evaluation of model quality. In this paper, we introduce a notion for the precision and fitness of an object-centric Petri net with respect to an object-centric event log. We give a formal definition and accompany this with an example. Furthermore, we provide an algorithm to calculate these quality measures. We discuss our precision and fitness notion based on an event log with different models. Our precision and fitness notions are an appropriate way to generalize quality measures to the object-centric setting since we are able to consider multiple case notions, their dependencies and their interactions.