Reachability in Geometrically $d$-Dimensional VASS
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915247176024064 |
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| author | Fu, Yuxi Zheng, Yangluo Yang, Qizhe |
| author_facet | Fu, Yuxi Zheng, Yangluo Yang, Qizhe |
| contents | 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 $d>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12302 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reachability in Geometrically $d$-Dimensional VASS Fu, Yuxi Zheng, Yangluo Yang, Qizhe Computational Complexity Formal Languages and Automata Theory Logic in Computer Science 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 $d>2$. |
| title | Reachability in Geometrically $d$-Dimensional VASS |
| topic | Computational Complexity Formal Languages and Automata Theory Logic in Computer Science |
| url | https://arxiv.org/abs/2504.12302 |