  Vector Addition Systems (VAS), aka Petri nets, are a popular model of
concurrency. The reachability set of a VAS is the set of configurations
reachable from the initial configuration. Leroux has studied the geometric
properties of VAS reachability sets, and used them to derive decision
procedures for important analysis problems. In this paper we continue the
geometric study of reachability sets. We show that every reachability set
admits a finite decomposition into disjoint almost hybridlinear sets enjoying
nice geometric properties. Further, we prove that the decomposition of the
reachability set of a given VAS is effectively computable. As a corollary, we
derive a simple algorithm for deciding semilinearity of VAS reachability sets,
the first one since Hauschildt's 1990 algorithm. As a second corollary, we
prove that the complement of a reachability set always contains an infinite
linear set.
