Petri Net Reachability Checking Is Polynomial with Optimal Abstraction Hierarchies

The Petri net model is a powerful state transition oriented model to analyse, model and evaluate asynchronous and concurrent systems. However, like other state transition models, it encounters the state explosion problem. The size of the state space increases exponentially with the system complexity. This paper is concerned with a method of abstracting automatically Petri nets to simpler representations, which are ordered with respect to their size. Thus it becomes possible to check Petri net reachability incrementally. With incremental approach we can overcome the exponential nature of Petri net reachability checking. We show that by using the incremental approach, the upper computational complexity bound for Petri net reachability checking with optimal abstraction hierarchies is polynomial. The method we propose considers structural properties of a Petri net as well an initial and a final marking. In addition to Petri net abstraction irrelevant transitions for a given reachability problem are determined. By removing these transitions from a net, impact of the state explosion problem is reduced even more.