Improving Reachability in Vector Addition Systems through Pumpability

Abstract: 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.

• The paper refines pumpability analysis for VAS, reducing the general reachability complexity from F_d to F_{d-2} and narrowing bounds compared to traditional VASS results.
• It employs geometric projection and cycle space techniques to establish PSPACE and ELEMENTARY results for 4-VAS and 5-VAS respectively.
• The improved method leads to shorter run witnesses and practical implications for verifying concurrent systems by leveraging the key parameter of geometric dimension.

Improving Reachability in Vector Addition Systems through Pumpability

Introduction and Motivation

The paper "Improving Reachability in Vector Addition Systems through Pumpability" (2604.24095) addresses algorithmic complexity in the reachability problem for vector addition systems (VAS), a fundamental model equiexpressive to Petri nets. Reachability in VAS forms the basis of numerous verification techniques for concurrent systems, but the computational complexity remains formidable—Ackermann-complete in general, and with significant gaps for fixed dimensions. Traditionally, upper bounds on the reachability problem in $d$-dimensional vector addition systems with states (VASS) have been inherited by VAS, despite the latter having reduced control structure. The main contribution of this work is a refined analysis of pumpability, leading to tighter upper bounds for VAS reachability.

Pumpability Refinement and Complexity Bounds

The technical heart of the work is a sharpened pumpability analysis for VAS that builds on Rackoff's extraction originally developed for VASS [DBLP:journals/tcs/Rackoff78, DBLP:conf/lics/LerouxS19]. The authors show that, for dimension $d$, the reachability problem in VAS can be solved in $\textsf{F}_{d-2}$-time, which improves upon the $\textsf{F}_d$-time bound previously derived for VASS. The fast-growing complexity hierarchy ($\textsf{F}_d$) is central to this work; it stratifies algorithmic problems well beyond elementary complexity, and previous upper bounds for VAS were loosely aligned with VASS complexity [DBLP:journals/toct/Schmitz16].

The pumpability criterion is formulated as follows: if all but one coordinates of a run in a wide sequential VAS reach sufficiently large values, then the target configuration is pumpable (i.e., can be incremented unboundedly in all coordinates, given suitable cycles). When pumpability fails, Rackoff’s dimension reduction applies, but the enhanced result allows for two bounded coordinates, not just one. Consequently, this enables the reduction of the $(d+2)$-VAS reachability problem to VASSes with geometric dimension at most $d$. This geometric dimension parameter (the span of cycle effects in strongly connected components) is increasingly critical in recent complexity analyses (Czerwiński et al., 17 Feb 2026).

Low-dimensional VAS: Improved Bounds and Techniques

Low-dimensional cases are examined in greater depth. For 4-VAS and 5-VAS, the paper establishes that reachability is in $\textsf{PSPACE}$ for 4-VAS and in $\textsf{ELEMENTARY}$ for 5-VAS—sometimes matching or improving bounds that were known for 2-VASS/3-VASS, but not previously for higher VAS dimensions. These results hinge on a novel projection technique for geometrically 2-dimensional VASS. The analysis shows that the reachability problem for these systems can be translated into equivalent 2-VASS reachability, enabling use of sharp bounds proven in prior works [DBLP:journals/jacm/BlondinEFGHLMT21, DBLP:conf/icalp/CzerwinskiJ0O25, DBLP:conf/icalp/FuYZ24, DBLP:conf/concur/Zheng25].

Specifically, for 4-VAS under unary encoding, reachability is shown to be -complete, and for geometrically 2-dimensional VASS (with dimension fixed), the reachability problem is also -complete. These results rely on careful decomposition and geometric projections, reducing the original problem to reachability in much lower dimensional systems.

Technical Insights and Numerical Evidence

The improved upper bounds are achieved via several technical mechanisms:

• Enhanced pumpability extraction: The technique ensures configurations with $d-1$ unbounded counters yield pumpability, facilitating shorter run witnesses and allowing dimension reduction to $d$0.
• Control of boundedness and geometric dimension: Two bounded coordinates in runs imply reduction to VASS with geometric dimension at most $d$1, allowing for finer-grained complexity analysis.
• Projection for low-dimensional systems: For 4-VAS and 5-VAS, the projection to lower geometric dimensions leverages semilinearity results and cycle space arguments for further reduction.

The paper offers strong numerical results. For example, shortest run witness lengths in 5-VAS are bounded by triply-exponential functions, and polynomial space suffices for 4-VAS reachability in binary encoding. In addition, reachability sets in 5-VAS are proven to be effectively semilinear.

Implications and Future Directions

Practically, these results refine the landscape for verification of concurrent systems modeled by VAS/Petri nets of fixed dimension, potentially enabling more scalable algorithms for low-dimensional reachability. Theoretically, the advances in pumpability extraction provide a blueprint for further complexity reductions in systems where control and dimension interact.

Open questions remain, such as whether run lengths in $d$2-VAS can be captured by $d$3-VASS, as well as tighter bounds for 5-VAS. Another challenge is the complexity gap in binary-encoded 2-VAS and 3-VASS, and the hardness for unary reachability in 2-VAS when dimension is not fixed. Refined analysis of geometric constraints in higher-dimensional systems is suggested as a promising direction, with the geometric dimension parameter offering continued leverage for both algorithmic and structural results.

Conclusion

The paper elucidates a refined pumpability analysis, substantially narrowing complexity upper bounds for reachability in fixed-dimensional VAS compared to prior VASS-derived results. The methodology is rigorous, combining geometric decomposition, cycle space analysis, and projections. The results have immediate implications for both the theoretical study and practical verification of concurrent systems, highlighting geometric dimension as a key tractability parameter and opening new pathways for analyses of low-dimensional vector addition systems.