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Languages of Boundedly-Ambiguous Vector Addition Systems with States

Authors Wojciech Czerwiński , Łukasz Orlikowski

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ACM Subject Classification
  • Theory of computation → Parallel computing models
Keywords
  • vector addition systems
  • Petri nets
  • unambiguity
  • bounded-ambiguity
  • languages

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Abstract

The aim of this paper is to deliver broad understanding of a class of languages of boundedly-ambiguous VASSs, that is k-ambiguous VASSs for some natural k. These are languages of Vector Addition Systems with States with the acceptance condition defined by the set of accepting states such that each accepted word has at most k accepting runs. We develop tools for proving that a given language is not accepted by any k-ambiguous VASS. Using them we show a few negative results: lack of some closure properties of languages of k-ambiguous VASSs and undecidability of the k-ambiguity problem, namely the question whether a given VASS language is a language of some k-ambiguous VASS. In fact we show an even more general undecidability result stating that for any class containing all regular languages and only k-ambiguous VASS languages for some k ∈ ℕ it is undecidable whether a language of a given 1-dimensional VASS belongs to this class. Finally, we show that the regularity problem is decidable for k-ambiguous VASSs.

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Wojciech Czerwiński and Łukasz Orlikowski. Languages of Boundedly-Ambiguous Vector Addition Systems with States. In 36th International Conference on Concurrency Theory (CONCUR 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 348, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CONCUR.2025.13

Author Details

Wojciech Czerwiński
  • University of Warsaw, Poland
Łukasz Orlikowski
  • University of Warsaw, Poland

Funding

Both authors are supported by the ERC grant INFSYS, agreement no. 950398.

Acknowledgements

We would like to thank David Purser for sharing with us a conjecture about language not recognised by an unambiguous VASS. We thank also Sławomir Lasota for suggesting us to reduce from 0-finite-reach instead of reducing from the halting problem for 2CM in the proof of Theorem 1. Additionally, we would like to thank Piotr Hofman for inspiring discussions and for reviewing the Master’s Thesis [Łukasz Orlikowski, 2024], which served as the foundation for the results presented in this paper.

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