Analysis of Petri Nets by Stepwise Refinements

Petri nets are being used by researchers in various domains where the notions of choice and concurrency are essential. Among these domains are the analysis of production schemata [3], the description and realization of digital systems [8], the finding of the computation rate of activities in asynchronous concurrent systems [l 11, the formal verification of parallel programs [6]. Many other applications can be imagined [IO]. Some interesting concepts such as boundedness, safeness and liveness have been defined on Petri nets [2], the analysis of a large Petri net is generally cumbersome or even impracticable. Another problem results from the fact that the firing of a transition is generally supposed non divisible and instantaneous. In fact when a Petri net is used to represent the functioning of an actual system, actions or operations are associated with the firing of the transitions. These actions or operations not being instantaneous, it is thus necessary to decompose each of them into a sequence ‘beginning of operation,” “end of operation”. The system has then to be depicted directly in detail and such a method does not allow stepwise refinements. It seems more efficient to suppose that the firing of a transition is not instantaenous and that it is made up of two steps [I 11. It is then possible to associate with a transition a complex operation that can later be depicted in detail by means of another Petri net. The aim of this paper is to prove that such a methodology allows a description and an analysis of a system by stepwise refinements. That is to say, it is possible to build up complex Petri nets with desired properties (bounded and live for example). An informal proof has been given in [12] and a more detailed approach of this problem can be found in [13].