Improved Ackermannian lower bound for the Petri nets reachability problem

Petri nets, equivalently presentable as vector addition systems with states, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there exists a sequence of valid execution steps reaching a given final configuration, is the central algorithmic problem for this model. The complexity of the problem has remained, until recently, one of the hardest open questions in verification of concurrent systems. A first upper bound has been provided only in 2015 by Leroux and Schmitz, then refined by the same authors to non-primitive recursive Ackermannian upper bound in 2019. The exponential space lower bound, shown by Lipton already in 1976, remained the only known for over 40 years until a breakthrough non-elementary lower bound by Czerwi{\'n}ski, Lasota, Lazic, Leroux and Mazowiecki in 2019. Finally, a matching Ackermannian lower bound announced this year by Czerwi{\'n}ski and Orlikowski, and independently by Leroux, established the complexity of the problem. Our contribution is an improvement of the former construction, making it conceptually simpler and more direct. On the way we improve the lower bound for vector addition systems with states in fixed dimension (or, equivalently, Petri nets with fixed number of places): while Czerwi{\'n}ski and Orlikowski prove $F_k$-hardness (hardness for $k$th level in Grzegorczyk Hierarchy) in dimension $6k$, and Leroux in dimension $4k+5$, our simplified construction yields $F_k$-hardness already in dimension $3k+2$.