Reachability in Fixed Dimension Vector Addition Systems with States

Authors Wojciech Czerwiński , Sławomir Lasota , Ranko Lazić , Jérôme Leroux, Filip Mazowiecki

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Author Details

Wojciech Czerwiński
  • University of Warsaw, Poland
Sławomir Lasota
  • University of Warsaw, Poland
Ranko Lazić
  • University of Warwick, Coventry, UK
Jérôme Leroux
  • CNRS & University of Bordeaux, France
Filip Mazowiecki
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany


We thank Matthias Englert for inspiring conversations.

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Wojciech Czerwiński, Sławomir Lasota, Ranko Lazić, Jérôme Leroux, and Filip Mazowiecki. Reachability in Fixed Dimension Vector Addition Systems with States. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The reachability problem is a central decision problem in verification of vector addition systems with states (VASS). In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability. We obtain three main results for VASS of fixed dimension. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension 3 whose shortest runs are exponential, and we show that the reachability problem is NP-hard in dimension 7. These results resolve negatively questions that had been posed by the works of Blondin et al. in LICS 2015 and Englert et al. in LICS 2016, and contribute a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional ones. Our third result, by means of a novel family of products of integer fractions, shows that 4-dimensional VASS can have doubly exponentially long shortest runs. The smallest dimension for which this was previously known is 14.

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Verification by model checking
  • Theory of computation → Logic and verification
  • reachability problem
  • vector addition systems
  • Petri nets


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