New Lower Bounds for Reachability in Vector Addition Systems

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

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Author Details

Wojciech Czerwiński
  • University of Warsaw, Poland
Ismaël Jecker
  • University of Warsaw, Poland
  • FEMTO-ST, CNRS, Univ. Franche-Comté, France
Sławomir Lasota
  • University of Warsaw, Poland
Jérôme Leroux
  • LaBRI, CNRS, Univ. Bordeaux, France
Łukasz Orlikowski
  • University of Warsaw, Poland


We are grateful to Weijun Chen for his sharp observation that brought to light a technical error in the initial version of this paper.

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Wojciech Czerwiński, Ismaël Jecker, Sławomir Lasota, Jérôme Leroux, and Łukasz Orlikowski. New Lower Bounds for Reachability in Vector Addition Systems. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We investigate the dimension-parametric complexity of the reachability problem in vector addition systems with states (VASS) and its extension with pushdown stack (pushdown VASS). Up to now, the problem is known to be F_d-hard for VASS of dimension 3d+2 (the complexity class F_d corresponds to the kth level of the fast-growing hierarchy), and no essentially better bound is known for pushdown VASS. We provide a new construction that improves the lower bound for VASS: F_d-hardness in dimension 2d+3. Furthermore, building on our new insights we show a new lower bound for pushdown VASS: F_d-hardness in dimension d/2 + 6. This dimension-parametric lower bound is strictly stronger than the upper bound for VASS, which suggests that the (still unknown) complexity of the reachability problem in pushdown VASS is higher than in plain VASS (where it is Ackermann-complete).

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Verification by model checking
  • Theory of computation → Logic and verification
  • vector addition systems
  • reachability problem
  • pushdown vector addition system
  • lower bounds


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