Reachability in Geometrically $d$-Dimensional VASS

Yuxi Fu; Yangluo Zheng; Qizhe Yang

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arXiv · 2504.12302

Reachability in Geometrically $d$-Dimensional VASS

Yuxi Fu·Yangluo Zheng·Qizhe Yang

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Abstract

Reachability of vector addition systems with states (VASS) is Ackermann complete~\cite{leroux2021reachability,czerwinski2021reachability}. For $d$-dimensional VASS reachability it is known that the problem is NP-complete~\cite{HaaseKreutzerOuaknineWorrell2009} when $d=1$, PSPACE-complete~\cite{BlondinFinkelGoellerHaaseMcKenzie2015} when $d=2$, and in $\mathbf{F}_d$~\cite{FuYangZheng2024} when $d>2$. A geometrically $d$-dimensional VASS is a $D$-dimensional VASS for some $D\ge d$ such that the space spanned by the displacements of the circular paths admitted in the $D$-dimensional VASS is $d$-dimensional. It is proved that the $\mathbf{F}_d$ upper bounds remain valid for the reachability problem in the geometrically $d$-dimensional VASSes with $d>2$.

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Yuxi Fu, Yangluo Zheng, Qizhe Yang. 2025-03-05. Reachability in Geometrically $d$-Dimensional VASS. https://arxiv.org/abs/2504.12302

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