Avoiding Exponential Explosion in Petri Net Models of Control Flows

Abstract

We look at modelling of a choice between several ‘bursts’ of concurrent actions in a Petri net. If ‘silent’ transitions are disallowed, a construction based on Cartesian product is traditionally used, resulting in an exponential explosion in the model size. We demonstrate that this exponential explosion can be avoided. We show the equivalence between this modelling problem and the problem of finding an edge clique cover of a complete multipartite graph, which gives major insights into the former problem as well as linking it to the existing results from graph theory. It turns out that the exponential number of places created by the Cartesian product construction can be improved down to polynomial (quadratic) in the worst case, and down to logarithmic in the best (non-degraded) case. For example, to express a choice between 10 pairs of concurrent transitions, the Cartesian product construction creates 1024 places, even though 6 places are sufficient. We also derive several lower and upper bounds on the numbers of places and arcs. As these results affect the ‘core’ modelling techniques based on Petri nets, eliminating a source of an exponential explosion, we hope they will have applications in Petri net modelling and translations of various formalisms to Petri nets. As an example, applying them to translate Burst Automata to Petri nets reduces the size of the resulting Petri net from exponential down to polynomial.