Reachability in 3-VASS is Elementary

TL;DR

This paper proves that the reachability problem in 3-dimensional vector addition systems with states (3-VASS) can be solved within doubly-exponential space, significantly narrowing the known complexity gap.

Contribution

It introduces a new upper bound on shortest path length in 3-VASS and a novel technique for approximating 2-VASS reachability sets using small semi-linear sets.

Findings

Reachability in 3-VASS is in doubly-exponential space

Shortest path length in 3-VASS is at most triply-exponential

New approximation technique for 2-VASS reachability sets

Abstract

The reachability problem in 3-dimensional vector addition systems with states (3-VASS) is known to be PSpace-hard, and to belong to Tower. We significantly narrow down the complexity gap by proving the problem to be solvable in doubly-exponential space. The result follows from a new upper bound on the length of the shortest path: if there is a path between two configurations of a 3-VASS then there is also one of at most triply-exponential length. We show it by introducing a novel technique of approximating the reachability sets of 2-VASS by small semi-linear sets.