Yangluo Zheng

Weijun Chen, Bo Fu, Yuxi Fu et al.

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.

Weijun Chen, Yuxi Fu, Yangluo Zheng

Vector addition systems (VAS) constitute an important model of computation and concurrency that is equally expressive as the Petri net model. Recently, a lot of research has been conducted on vector addition systems with states (VASS), which are VASes equipped with a finite state control. Results on VASS naturally carry over to VAS, but no straightforward improvement is available. In this paper, we investigate the reachability problem in VAS in fixed dimensions. Based on a pumpability analysis of VAS that refines Rackoff's extraction for VASS, we obtain an F_{d-2} upper bound for the d-dimensional VAS reachability problem, improving the F_d upper bound inherited from the d-dimensional VASS reachability problem. Low-dimensional VASes are also considered. In particular, we establish a PSPACE upper bound for reachability in 4-dimensional VAS and an ELEMENTARY upper bound for 5-dimensional VAS, while the same upper bounds were known only for 2-VASS and 3-VASS, respectively. The result for 4-VAS particularly hinges on a simplified projection technique developed for geometrically 2-dimensional VASSes, whose reachability problem is shown to be equivalent to 2-VASS.