Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable

Authors Wojciech Czerwiński , Piotr Hofman

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Wojciech Czerwiński
  • University of Warsaw, Poland
Piotr Hofman
  • University of Warsaw, Poland


We thank Filip Mazowiecki for asking the question for boundedly-ambiguous VASSes and formulating the conjecture that control automata of boundedly-ambiguous VASSes can be made boundedly-ambiguous. We also thank him and David Purser for inspiring discussions on the problem. We thank Thomas Colcombet for suggesting the way of proving Theorem 25, Mahsa Shirmohammadi for pointing us to the undecidability result [Petr Jancar, 2001] and Lorenzo Clemente for inspiring discussions on weighted models.

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Wojciech Czerwiński and Piotr Hofman. Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parallel computing models
  • vector addition systems
  • language inclusion
  • language equivalence
  • determinism
  • unambiguity
  • bounded ambiguity
  • Petri nets
  • well-structured transition systems


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