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Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality

Authors Marvin Künnemann, Filip Mazowiecki, Lia Schütze , Henry Sinclair-Banks , Karol Węgrzycki

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  • Theory of computation → Models of computation
Keywords
  • Vector Addition System
  • Coverability
  • Reachability
  • Fine-Grained Complexity
  • Exponential Time Hypothesis
  • k-Cycle Hypothesis
  • Hyperclique Hypothesis

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Abstract

Seminal results establish that the coverability problem for Vector Addition Systems with States (VASS) is in EXPSPACE (Rackoff, '78) and is EXPSPACE-hard already under unary encodings (Lipton, '76). More precisely, Rosier and Yen later utilise Rackoff’s bounding technique to show that if coverability holds then there is a run of length at most n^{2^𝒪(d log d)}, where d is the dimension and n is the size of the given unary VASS. Earlier, Lipton showed that there exist instances of coverability in d-dimensional unary VASS that are only witnessed by runs of length at least n^{2^Ω(d)}. Our first result closes this gap. We improve the upper bound by removing the twice-exponentiated log(d) factor, thus matching Lipton’s lower bound. This closes the corresponding gap for the exact space required to decide coverability. This also yields a deterministic n^{2^𝒪(d)}-time algorithm for coverability. Our second result is a matching lower bound, that there does not exist a deterministic n^{2^o(d)}-time algorithm, conditioned upon the Exponential Time Hypothesis.
When analysing coverability, a standard proof technique is to consider VASS with bounded counters. Bounded VASS make for an interesting and popular model due to strong connections with timed automata. Withal, we study a natural setting where the counter bound is linear in the size of the VASS. Here the trivial exhaustive search algorithm runs in 𝒪(n^{d+1})-time. We give evidence to this being near-optimal. We prove that in dimension one this trivial algorithm is conditionally optimal, by showing that n^{2-o(1)}-time is required under the k-cycle hypothesis. In general fixed dimension d, we show that n^{d-2-o(1)}-time is required under the 3-uniform hyperclique hypothesis.

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Marvin Künnemann, Filip Mazowiecki, Lia Schütze, Henry Sinclair-Banks, and Karol Węgrzycki. Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 131:1-131:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.131

Author Details

Marvin Künnemann
  • RPTU Kaiserslautern-Landau, Germany
Filip Mazowiecki
  • University of Warsaw, Poland
Lia Schütze
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Henry Sinclair-Banks
  • Centre for Discrete Mathematics and its Applications (DIMAP) & Department of Computer Science, University of Warwick, Coventry, UK
Karol Węgrzycki
  • Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Künnemann, Marvin: Research partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 462679611.
  • Mazowiecki, Filip: Supported by the ERC grant INFSYS, agreement no. 950398.
  • Sinclair-Banks, Henry: Supported by EPSRC Standard Research Studentship (DTP), grant number EP/T5179X/1.
  • Węgrzycki, Karol: Supported by the ERC grant TIPEA, agreement no. 850979.

Acknowledgements

We would like to thank our anonymous reviewers for their comments, for their time, and especially for highlighting a piece of related work [Ulla Koppenhagen and Ernst W. Mayr, 2000].

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