New Lower Bounds for Reachability in Vector Addition Systems

TL;DR

This paper improves the known lower bounds for the complexity of the reachability problem in vector addition systems and pushdown VASS, indicating higher complexity than previously established and suggesting the problem may be harder than Ackermann-complete.

Contribution

It provides new dimension-parametric lower bounds for the reachability problem in VASS and pushdown VASS, surpassing previous bounds and indicating increased complexity.

Findings

Lower bound for VASS: $ ext{F}_k$-hardness in dimension $2k+3$

Lower bound for pushdown VASS: $ ext{F}_k$-hardness in dimension $rac{k}{2}+4$

Complexity of pushdown VASS likely exceeds Ackermann complexity

Abstract

We investigate the dimension-parametric complexity of the reachability problem in vector addition systems with states (VASS) and its extension with pushdown stack (pushdown VASS). Up to now, the problem is known to be $\mathcal{F}_k$-hard for VASS of dimension $3k+2$ (the complexity class $\mathcal{F}_k$ corresponds to the $k$th level of the fast-growing hierarchy), and no essentially better bound is known for pushdown VASS. We provide a new construction that improves the lower bound for VASS: $\mathcal{F}_k$-hardness in dimension $2k+3$. Furthermore, building on our new insights we show a new lower bound for pushdown VASS: $\mathcal{F}_k$-hardness in dimension $\frac k 2 + 4$. This dimension-parametric lower bound is strictly stronger than the upper bound for VASS, which suggests that the (still unknown) complexity of the reachability problem in pushdown VASS is higher than in plain VASS (where it is Ackermann-complete).