Van der Aalst’s theorem is an important result for the analysis and synthesis of process models. The paper proves the theorem by exhausting perpetual free-choice Petri nets by CP-subnets. The resulting T-systems are investigated by elementary methods.

Bipolar synchronization systems (BP-systems) constitute a class of coloured Petri nets, well suited for modelling the control flow of discrete dynamical systems. Every BP-system has an underlying ordinary Petri net, a T-system. It further has a second ordinary net attached, a free-choice system. We prove that a BP-system is safe and live if the T-system and the free-choice system are safe and live and the free-choice system in addition has no frozen tokens. This result is the converse of a theorem of Genrich and Thiagarajan and proves an old conjecture. As a consequence we obtain two results about the existence of safe and live BP-systems with prescribed ordinary Petri nets. For the proof of these theorems we introduce the concept of a morphism between Petri nets as a means of comparing different Petri nets. We then apply the classical theory of free-choice systems.