Rüdiger Valk

Petri Nets as Token Objects

The model of Elementary Object System is introduced and motivated by several examples and applications. Object systems support a modeling of systems by Petri nets following the paradigm of Object-Oriented Modeling. They are composed of a System Net and one or more Object Nets which can be seen as token objects of the system net. By this approach an interesting and challenging two-level system modeling technique is introduced. Similar to the object-oriented approach, complex systems are modeled close to their real appearance in a natural way to promote clear and reliable concepts. Applications in fields like work-flow, flexible manufacturing or agent-oriented approaches (mobile agents and/or intelligent agents as in AI research) are feasible. This paper gives an introduction with several examples, but only few definitions and no theorems, which can be found, however, in a more elaborated paper [19].

Petri nets and regular languages

It is shown that the regularity problem for firing sequence sets of Petri nets is decidable. For the proof, new techniques to characterize unbounded places are introduced. In the class L0 of terminal languages of labelled Petri nets the regularity problem in undecidable. In addition some lower bounds for the undecidability of the equality problems in L0 and L are given. L0λ is shown to be not closed under complementation without reference to the reachability problem.

The residue of vector sets with applications to decidability problems in Petri nets

A set K of integer vectors is called right-closed, if for any elementmeK all vectors m′≧m are also contained in K. In such a case K is a semilinear set of vectors having a minimal generating set res(K), called the residue of K. A general method is given for computing the residue set of a right-closed set, provided it satisfies a certain decidability criterion.

Concurrency in communicating object petri nets

Objects are studied as higher-level net tokens having an individual dynamical behaviour. In the context of Petri net research it i s quite natural to also model such tokens by Petri nets. To distinguish them from the system net, they are called object nets. Object nets behave like tokens, i.e., they are lying in places and are moved by transitions. In contrast to ordinary tokens, however, they may change their state (i.e. their marking) when lying in a place or when being moved by a transition. By this approach an interesting and challenging two-level system modelling technique is introduced. Similar to the object-oriented approach, complex systems are modelled close to their real appearance in a natural way to promote clear and reliable concepts. Applications in fields like workflow, agent-oriented approaches (mobile agents and/or intelligent agents as in AI research) or open system networks are feasible. This paper gives a precise definition of the basic model together with a suitable process semantics. The focus is set more on basic concepts and their fundamental study than on high modelling capability.

Object Petri Nets

The nets-within-nets paradigm provides an innovative mod- elling technique by giving tokens themselves the structure of a Petri net. These nets, called token nets or object nets, also support the object ori- ented modelling technique as they may represent real world objects with a proper dynamical behaviour. Between object nets and the surrounding net, called system net, various interaction mechanisms exist as well as between different object nets. This introduction into the field of object Petri nets starts with small examples and proceeds by giving formal se- mantics. Some of the examples are modelled within the formalism of the Renew tool. Finally the differences between reference and two kinds of value semantics are discussed.

Infinite behaviour and fairness in Petri nets

Several classes of ω-languages of labelled Petri nets are defined and related to each other. Since such nets can be interpreted to behave fair, these notions are compared with explicit definitions of fairness for nets.

Nets in Computer Organisation

It is shown how net theory can be used for modelling task flow in systems of functional units. By an example also time depending considerations are presented, as usually done in performance evaluation.

Infinite Behaviour and Fairness

Specification techniques for the infinite behaviour of a P/T-system are introduced and their expressive power is compared. In particular, fair behaviour is related to the wellknown liveness problem. Finally, the problem is studied how to implement fair behaviour using a fair occurrence rule for transitions.

Object Oriented Petri Nets in Business Process Modeling

Business systems have to adapt to changing requirements coming from their environment. The rate is continuously increasing and leads to massive use of computer based systems. To specify the systems in a way that allows for adaptability and flexibility adequate techniques are necessary. The disadvantages using traditional modeling techniques are partially overcome by Business Process Petri nets (BPP-nets) which are informally introduced in this contribution. The key concepts are an object oriented structure of the net models, allowing to partition the model according to an application and also to follow a process centered approach. Workflows within the system can be modeled in separate objects and thus allow the dynamic adaptation of the system if the environment requires a behavioral change.