Box-Reachability in Vector Addition Systems

TL;DR

This paper introduces and analyzes box-reachability in Vector Addition Systems, showing that in 2D, the box-reachable set closely approximates the standard reachability set beyond a certain threshold, using convex geometry techniques.

Contribution

It defines box-reachability in VAS and proves that in 2D, the box-reachable set nearly matches the standard reachability set above a threshold, advancing understanding of VAS reachability.

Findings

In 2D VAS, box-reachability set coincides with standard reachability set beyond a threshold W.

Box-reachability shares properties with standard reachability, with notable differences.

Convex geometry tools are used to establish the main results.

Abstract

We consider a variant of reachability in Vector Addition Systems (VAS) dubbed \emph{box reachability}, whereby a vector $v\in \mathbb{N}^d$ 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.