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Reachability in 3-VASS Is Elementary

Authors Wojciech Czerwiński , Ismaël Jecker , Sławomir Lasota , Łukasz Orlikowski

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  • Theory of computation → Parallel computing models
Keywords
  • vector addition systems
  • Petri nets
  • reachability problem
  • dimension three
  • doubly exponential space
  • length of shortest path

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Abstract

The reachability problem in 3-dimensional vector addition systems with states (3-VASS) is known to be PSpace-hard, and to belong to Tower. We significantly narrow down the complexity gap by proving the problem to be solvable in doubly-exponential space. The result follows from a new upper bound on the length of the shortest path: if there is a path between two configurations of a 3-VASS then there is also one of at most triply-exponential length. We show it by introducing a novel technique of approximating the reachability sets of 2-VASS by small semi-linear sets.

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Wojciech Czerwiński, Ismaël Jecker, Sławomir Lasota, and Łukasz Orlikowski. Reachability in 3-VASS Is Elementary. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 153:1-153:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.153

Author Details

Wojciech Czerwiński
  • University of Warsaw, Poland
Ismaël Jecker
  • Université Marie et Louis Pasteur, CNRS, institut FEMTO-ST, F-25000 Besançon, France
Sławomir Lasota
  • University of Warsaw, Poland
Łukasz Orlikowski
  • University of Warsaw, Poland

Funding

  • Czerwiński, Wojciech: Supported by the ERC grant INFSYS, agreement no. 950398.
  • Lasota, Sławomir: Supported by the ERC grant INFSYS, agreement no. 950398 and by the NCN grant 2021/41/B/ST6/00535.
  • Orlikowski, Łukasz: Supported by the ERC grant INFSYS, agreement no. 950398.

Acknowledgements

We thank Radosław Piórkowski for many inspiring discussions about cones in VASS a few years ago and Filip Mazowiecki for the discussions about the reachability problem for #1-ǎss 3. We thank also Henry Sinclair-Banks for helpful discussions about [Dmitry Chistikov et al., 2024]. We thank Qizhe Yang and Yangluo Zheng for the discussions on 3-VASS, which are geometrically 2-dimensional. Finally, we thank anonymous reviewers for helpful comments.

References

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