Separators in Continuous Petri Nets

Abstract

AbstractLeroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $$\boldsymbol{m}_\text {src}$$ m src cannot reach a marking $$\boldsymbol{m}_\text {tgt}$$ m tgt , then there is a formula $$\varphi $$ φ of Presburger arithmetic such that: $$\varphi (\boldsymbol{m}_\text {src})$$ φ ( m src ) holds; $$\varphi $$ φ is forward invariant, i.e., $$\varphi (\boldsymbol{m})$$ φ ( m ) and $$\boldsymbol{m} \rightarrow \boldsymbol{m}'$$ m → m ′ imply $$\varphi (\boldsymbol{m}'$$ φ ( m ′ ); and $$\lnot \varphi (\boldsymbol{m}_\text {tgt})$$ ¬ φ ( m tgt ) holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their (super-)Ackermannian worst-case size and the (super-)exponential complexity of checking that a formula is a separator. We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachablity, which is P-complete).