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arXiv · 2603.05221

Reachability in VASS Extended with Integer Counters

Clotilde Bizière·Wojciech Czerwiński·Roland Guttenberg·Jérôme Leroux·Vincent Michielini·Łukasz Orlikowski·Antoni Puch·Henry Sinclair-Banks
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Abstract

We consider a variant of VASS extended with integer counters, denoted VASS+Z. These are automata equipped with N and Z counters; the N-counters are required to remain nonnegative and the Z-counters do not have this restriction. We study the complexity of the reachability problem for VASS+Z when the number of N-counters is fixed. We show that reachability is NP-complete in 1-VASS+Z (i.e. when there is only one N-counter) regardless of unary or binary encoding. For $d \geq 2$, using a KLMST-based algorithm, we prove that reachability in d-VASS+Z lies in the complexity class $\mathcal{F}_{d+2}$. Our upper bound improves on the naively obtained Ackermannian complexity by simulating the Z-counters with N-counters. To complement our upper bounds, we show that extending VASS with integer counters significantly lowers the number of N-counters needed to exhibit hardness. We prove that reachability in unary 2-VASS+Z is PSPACE-hard; without Z-counters this lower bound is only known in dimension 5. We also prove that reachability in unary 3-VASS+Z is TOWER-hard. Without Z-counters, reachability in 3-VASS has elementary complexity and TOWER-hardness is only known in dimension 8.

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Clotilde Bizière, Wojciech Czerwiński, Roland Guttenberg, Jérôme Leroux, Vincent Michielini, Łukasz Orlikowski, Antoni Puch, Henry Sinclair-Banks. 2026-03-05. Reachability in VASS Extended with Integer Counters. https://doi.org/10.4230/lipics.lics.2026.19

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