Parametrized Universality Problems for One-Counter Nets

Authors Shaull Almagor , Udi Boker, Piotr Hofman , Patrick Totzke

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Shaull Almagor
  • Technion - Israel Institute of Technology, Haifa, Israel
Udi Boker
  • Interdisciplinary Center (IDC) Herzliya, Israel
Piotr Hofman
  • University of Warsaw, Poland
Patrick Totzke
  • University of Liverpool, UK


We are grateful for fruitful discussions during the Autoboz'2019 workshop.

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Shaull Almagor, Udi Boker, Piotr Hofman, and Patrick Totzke. Parametrized Universality Problems for One-Counter Nets. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the language universality problem for One-Counter Nets, also known as 1-dimensional Vector Addition Systems with States (1-VASS), parameterized either with an initial counter value, or with an upper bound on the allowed counter value during runs. The language accepted by an OCN (defined by reaching a final control state) is monotone in both parameters. This yields two natural questions: 1) does there exist an initial counter value that makes the language universal? 2) does there exist a sufficiently high ceiling so that the bounded language is universal? Although the ordinary universality problem is decidable (and Ackermann-complete) and these parameterized variants seem to reduce to checking basic structural properties of the underlying automaton, we show that in fact both problems are undecidable. We also look into the complexities of the problems for several decidable subclasses, namely for unambiguous, and deterministic systems, and for those over a single-letter alphabet.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Counter net
  • VASS
  • Unambiguous Automata
  • Universality


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