Reachability in 3-VASS is Elementary
Wojciech Czerwiński, Ismaël Jecker, Sławomir Lasota, Łukasz Orlikowski
TL;DR
This work establishes an elementary upper bound for reachability in 3-VASS by proving that the shortest path between configurations can be bounded by ${ extsc{size}}(V,s,t)^{2^{2^{ ext{O}(k)}}}$, enabling a 2-ExpSpace algorithm for binary-encoded instances. The authors introduce a novel technique to approximate 2-VASS reachability via small semi-linear sets, and develop the notion of polynomially approximable sets to control blowups when reducing k-component 3-VASS to smaller instances. The core proof proceeds by induction on the number of components, leveraging sequential cones and geometric decompositions to replace components with geometrically 2-dimensional VASS and apply the 2-VASS semi-linear machinery. Consequently, they obtain an elementary upper bound and fixed-k complexity results (NL/PSpace), significantly narrowing the prior gap between PSpace and Tower bounds and opening new avenues for approximation-based methods in reachability problems.
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.