Reachability in Five-Dimensional Vector Addition Systems Is PSPACE-Complete

Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.