On the Reachability Problem for Two-Dimensional Branching VASS

TL;DR

This paper proves that the reachability problem for two-dimensional branching VASS (BVASS) is decidable and that the reachability set can be represented as a computable semilinear set, advancing understanding of these systems.

Contribution

We establish the decidability of the reachability problem for 2D BVASS and show the reachability set admits a computable semilinear representation, a novel result for this model.

Findings

Reachability for 2D BVASS is decidable.

Reachability set is semilinear and computable.

Open problem remains in higher dimensions.

Abstract

Vectors addition systems with states (VASS), or equivalently Petri nets, are arguably one of the most studied formalisms for the modeling and analysis of concurrent systems. A central decision problem for VASS is reachability: whether there exists a run from an initial configuration to a final one. This problem has been known to be decidable for over forty years, and its complexity has recently been precisely characterized. Our work concerns the reachability problem for BVASS, a branching generalization of VASS. In dimension one, the exact complexity of this problem is known. In this paper, we prove that the reachability problem for 2-dimensional BVASS is decidable. In fact, we even show that the reachability set admits a computable semilinear presentation. The decidability status of the reachability problem for BVASS remains open in higher dimensions.