Reachability in Vector Addition Systems is Ackermann-complete

TL;DR

This paper proves that the reachability problem in Vector Addition Systems and Petri nets is Ackermann-complete, resolving a 45-year open problem and establishing its precise high complexity.

Contribution

It establishes the exact Ackermann-complete complexity of the reachability problem in Vector Addition Systems, closing a long-standing open question.

Findings

Reachability in Vector Addition Systems is Ackermann-complete.

The problem is $ ext{F}_k$-hard for systems with dimension $6k$.

Closes a 45-year open problem in the field.

Abstract

Vector Addition Systems and equivalent Petri nets are a well established models of concurrency. The central algorithmic problem for Vector Addition Systems with a long research history is the reachability problem asking whether there exists a run from one given configuration to another. We settle its complexity to be Ackermann-complete thus closing the problem open for 45 years. In particular we prove that the problem is $\mathcal{F}_k$-hard for Vector Addition Systems with States in dimension $6k$, where $\mathcal{F}_k$ is the $k$-th complexity class from the hierarchy of fast-growing complexity classes.