This is the introductory module to the Petri Net Course and as such provides key concepts and definitions underlying almost every Petri net model. Guided by a motivating example, principles of net theory are discussed highlighting local dynamics and concurrency. Two basic net classes are introduced and investigated: Place/Transition Systems and Elementary Net (EN) Systems. We consider the occurrence rule (token game), reachability, state graph, behavioural properties like deadlock and boundedness, behavioural equivalence and normal forms. The fundamental situations causality, conflict, concurrency, and confusion are explained in the context of EN Systems. We discuss EN system behaviour in terms of sequential and non-sequential observations. Finally, basic analysis techniques to establish structural properties of nets are presented.

This module focusses on the constructs and definition of the Coloured Petri Nets (CPNs) modelling language. CPNs belong to the class of high-level Petri nets and combines Petri Nets with the functional programming language Standard ML (SML). Petri nets provides the primitives for modelling concurrency, communication, and synchronisation while SML provides the primitives for modelling data manipulation and for creating compact and parameterised models. CPNs and the supporting computer tool CPN Tools have been widely used in practice for modelling and validating a wide range of concurrent and distributed systems. Examples demonstrating the practical use of CPN modelling and verification on industrial-sized systems will be presented. Having completed this module the participants should be able to:

The module includes hands-on experiments with CPN Tools.

In the beginning, model checking was just a set of algorithms: given a system model and a specification (written in a temporal logic), decide whether the model satisfies the specification. The problem is challenging, mainly due to the state explosion problem. State explosion can be addressed in various ways. This has led to a wealth of technology: data structures, implementations, and approaches. In this module, model checking differentiated by the application domain. For instance, the main challenge in software model checking is to find appropriate abstractions for the data structures.

Petri net model checking has developed into its own branch of model checking. It can be characterized by:

We cover the whole spectrum from basic algorithms to state-of-the-art technology. At every stage, we show where and how our application domain Petri nets impacts the design of a Petri net model checker. We demonstrate the results using the LoLA 2 model checking tool that is freely available.

This module presents different ways to introduce time in Petri nets, focusing on the various policies and ideas explored in literature. In particular, four model classes will be considered in details:

The module will introduce the syntax and semantics of these model classes, and develop some standard analysis techniques.

Furthermore, the module will include a short description of Markov chains in order to be self-contained.

Object-centricity has been among the key trends in the process science community in recent years. In a nutshell, object-centric processes must exhibit two key features: the ability to identify the objects they manipulate and the capacity to represent relations between such objects, where the relations may evolve during process execution. The first feature is typically achieved by introducing tokens with unique identities (similar to colored Petri nets), where the identifier domain consists of countably many elements. The second feature can be obtained in various ways. For instance, one approach involves using places to store tuples of related objects, while another relies on refined execution semantics that enable the simultaneous firing of multiple transitions, which in turn consume or transfer related tokens.

To date, various Petri net classes have been employed for modeling and analyzing object-centric processes. In this tutorial, we will provide a brief overview of these classes and focus on a specific one: typed Petri nets with identifiers (t-PNIDs for short). This class allows to distinguish objects not only by their identifiers but also by their types. Places in t-PNIDs represent relations (with fixed schemas) between objects which in turn means that tokens carry tuples of identifiers. Objects can participate in multiple relations and, throughout net executions, can “migrate” from one relation to another. Additionally, t-PNIDs support the generation of fresh identifiers, allowing for dynamic object creation.

In this tutorial, we will show how t-PNIDs can be used as a modelling tool for object-centric processes. Furthermore, we will discuss key properties of t-PNIDs and their decidability. Among such properties, specific attention will be put on the property of soundness, which is crucial for workflow-like models and plays a significant role in process discovery. In the final part of the tutorial, we will discuss how t-PNIDs (and their sub-classes) can be used in various process mining tasks.

new content

The deadline for submitting the assignments is August 31, 2026. Please submit your assignments by email to the respective instructors responsible for the tasks. The assignments can be found both in the respective module folders and in the “Assignments” folder.