The Reachability Problem for Petri Nets is Not Primitive Recursive

TL;DR

This paper establishes that the reachability problem for Petri nets has extremely high computational complexity, specifically Ackermannian, and is not elementary even in fixed dimensions, highlighting fundamental computational limits.

Contribution

It proves a new Ackermannian lower bound for Petri net reachability and shows non-elementary complexity in fixed dimensions, advancing understanding of the problem's computational hardness.

Findings

Reachability problem has Ackermannian complexity lower bound.

In fixed dimension 2d+4, the problem is -hard.

In dimension 10, the problem is not elementary.

Abstract

We provide an Ackermannian complexity lower bound for the reachability problem for checking programs, a model equivalent to Petri nets. Moreover in fixed dimension $2d+4$, we show that the problem is $\mathbb{F}_d$-hard. As a direct corollary, the reachability problem in dimension 10 is not elementary.