Geometry of VAS reachability sets

Vector Addition Systems (VAS) or equivalently petri-nets are a popular model for representing concurrent systems. Many important decidability results about VAS were obtained by considering geometric properties of their reachability sets, i.e. the set of configurations reachable from some initial configuration $c_0$. For example, in 2012 Jerome Leroux proved that if a configuration $c_t$ is not reachable, then there exists a semilinear inductive invariant separating the reachability set from $c_t$. This gave an alternative proof of decidability of the reachability problem. The paper introduced the class of petri-sets, proved that reachability sets are petri-sets, and that petri-sets have this property. In a follow-up paper in 2013, Jerome Leroux again used the class of petri-sets to prove that if a reachability set is semilinear, then a representation of it can be computed. In this paper, we utilize the class of petri-sets to answer the opposite type of question: Even if the reachability set is non-semilinear, what form can it have? We give another proof that semilinearity of the reachability set is decidable, which was first shown by Hauschildt in 1990. We prove that reachability sets can be partitioned into nicely shaped sets we call almost-hybridlinear, and how to utilize this to decide semilinearity.