Abstract
Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking \(\boldsymbol{m}_\text {src}\) cannot reach a marking \(\boldsymbol{m}_\text {tgt}\), then there is a formula \(\varphi \) of Presburger arithmetic such that: \(\varphi (\boldsymbol{m}_\text {src})\) holds; \(\varphi \) is forward invariant, i.e., \(\varphi (\boldsymbol{m})\) and \(\boldsymbol{m} \rightarrow \boldsymbol{m}'\) imply \(\varphi (\boldsymbol{m}'\)); and \(\lnot \varphi (\boldsymbol{m}_\text {tgt})\) 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).
M. Blondin was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the Fonds de recherche du Québec – Nature et technologies (FRQNT). J. Esparza was supported by an ERC Advanced Grant (787367: PaVeS).
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Blondin, M., Esparza, J. (2022). Separators in Continuous Petri Nets. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_5
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