# Reachability in Vector Addition Systems is Primitive-Recursive in Fixed Dimension

## TL;DR

This paper proves that the reachability problem in fixed-dimension vector addition systems has a primitive-recursive upper bound, refining previous bounds and improving understanding of its computational complexity.

## Contribution

The authors refine classical algorithms to establish primitive-recursive bounds for fixed-dimension systems, advancing the theoretical understanding of reachability complexity.

## Key findings

- Achieved primitive-recursive upper bounds in fixed dimension
- Refined classical decomposition algorithms for vector addition systems
- Provided Ackermann upper bounds in the general case

## Abstract

The reachability problem in vector addition systems is a central question, not only for the static verification of these systems, but also for many inter-reducible decision problems occurring in various fields. The currently best known upper bound on this problem is not primitive-recursive, even when considering systems of fixed dimension. We provide significant refinements to the classical decomposition algorithm of Mayr, Kosaraju, and Lambert and to its termination proof, which yield an ACKERMANN upper bound in the general case, and primitive-recursive upper bounds in fixed dimension. While this does not match the currently best known TOWER lower bound for reachability, it is optimal for related problems.