Place Bisimilarity is Decidable, Indeed!
Roberto Gorrieri
TL;DR
It is shown that it is possible to decide $\sim_p$ with a simple algorithm, which essentially scans all the place relations to check whether they are place bisimulations, and it does respect the intended causal semantics of Petri nets, as it is finer than causal-net bisimilarity.
Abstract
Place bisimilarity $\sim_p$ is a behavioral equivalence for finite Petri nets, originally proposed in \cite{ABS91}, that, differently from all the other behavioral relations proposed so far, is not defined over the markings of a finite net, rather over its places, which are finitely many. Place bisimilarity $\sim_p$ was claimed decidable in \cite{ABS91}, but its decidability was not really proved. We show that it is possible to decide $\sim_p$ with a simple algorithm, which essentially scans all the place relations (which are finitely many) to check whether they are place bisimulations. We also show that $\sim_p$ does respect the intended causal semantics of Petri nets, as it is finer than causal-net bisimilarity \cite{Gor22}. Moreover, we propose a slightly coarser variant, we call d-place bisimilarity $\sim_d$, that we conjecture to be the coarsest equivalence, fully respecting causality and branching time (as it is finer than fully-concurrent bisimilarity \cite{BDKP91}), to be decidable on finite Petri nets. Finally, two even coarser variants are discussed, namely i-place and i-d-place bisimilarities, which are still decidable, do preserve the concurrent behavior of Petri nets, but do not respect causality. These results open the way towards formal verification (by equivalence checking) of distributed systems modeled by finite Petri nets.