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).
Publisher
Springer International Publishing