Bo Fu

Abstract. 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 ≥ 3. For 5-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed d ≥ 3, it also yields an F_d upper bound for reachability in (d+3)-VAS.

Abstract. A VASS can be viewed as a finite-state automaton manipulating a fixed number (called its dimension) of counters holding non-negative values. The reachability problem, asking whether there is a run from one configuration, defined by a state and values of the counters, to another configuration, has been a long-standing algorithmic challenge in theoretical computer science. When the dimension is part of the input, the problem has been shown to be ACKERMANN-complete in 2021.

For fixed dimension greater than 2, and in particular for dimension 3, the exact complexity of the reachability problem remains unclear. For a long time the known algorithms for the 3-dimensional VASS reachability problem had been non-elementary, while the best known lower bound is merely PSPACE hardness inherited from dimension 2. A recent breakthrough in (Czerwiński, Jecker, Lasota, Orlikowski, ICALP 2025) gave the first elementary upper bound for the problem, namely 2-EXPSPACE.

In this paper it is shown that the reachability problem in 3-VASS belongs to EXPSPACE. The proof is based on a hierarchical pumpability analysis, yielding a doubly-exponential length bound on the shortest runs between two configurations.