Box-Reachability in Vector Addition Systems

We consider a variant of reachability in Vector Addition Systems (VAS) dubbed box reachability, whereby a vector v∈ Nᵈ is box-reachable from $0$ in a VAS V if V admits a path from $0$ to v that not only stays in the positive orthant (as in the standard VAS semantics), but also stays below v, i.e., within the ``box'' whose opposite corners are $0$ and v. Our main result is that for two-dimensional VAS, the set of box-reachable vertices almost coincides with the standard reachability set: the two sets coincide for all vectors whose coordinates are both above some threshold W. We also study properties of box-reachability, exploring the differences and similarities with standard reachability. Technically, our main result is proved using powerful machinery from convex geometry.