On Cyclic Behaviour of Unbounded Petri Nets

Abstract

Cycles in state spaces represent repetitive behaviour of system models. Runs reproducing some state have important interpretations, for example rounds in distributed algorithms. In case of unbounded system models with infinite state space, cycles cannot be found in a straightforward way. For Petri nets, transition invariants provide necessary conditions for cyclic behaviour, but not for every transition invariant there is a corresponding cycle. Another approach to deal with infinite state behaviour is to consider finite coverability graphs which generalize reachability graphs by adding the value "arbitrary many" for unbounded places. Unfortunately, a cycle in the coverability graph does not necessarily represent a cyclic behaviour. This paper combines the concepts transition invariant and coverability graph in such a way that cyclic behaviour can be found in a combined graph. This implies a way to decide whether a sequence constitutes a cycle. A finite representation of all (infinitely many) cycles is implied by a result stating that the set of cycles is semi-linear. We also discuss an application of this concept: schedulability of Petri nets, i.e., control of transition occurrences such that the controlled behaviour does not lead to arbitrary token growth on any place.