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Two-Way One-Counter Nets Revisited

Authors Shaull Almagor , Michaël Cadilhac , Asaf Yeshurun

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Author Details

Shaull Almagor
  • Department of Computer Science, Technion, Haifa, Israel
Michaël Cadilhac
  • DePaul University, Chicago, IL, USA
Asaf Yeshurun
  • Department of Computer Science, Technion, Haifa, Israel

Funding

  • Almagor, Shaull: supported by the ISRAEL SCIENCE FOUNDATION (grant No. 989/22).

Acknowledgements

We are grateful to Dmitry Chistikov for shedding light on some claims made in [Arikawa, 1968] and to an anonymous reviewer for providing some key references.

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Shaull Almagor, Michaël Cadilhac, and Asaf Yeshurun. Two-Way One-Counter Nets Revisited. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.19

Abstract

One Counter Nets (OCNs) are finite-state automata equipped with a counter that cannot become negative, but cannot be explicitly tested for zero. Their close connection to various other models (e.g., PDAs, Vector Addition Systems, and Counter Automata) make them an attractive modeling tool.
The two-way variant of OCNs (2-OCNs) was introduced in the 1980’s and shown to be more expressive than OCNs, so much so that the emptiness problem is undecidable already in the deterministic model (2-DOCNs).
In a first part, we study the emptiness problem of natural restrictions of 2-OCNs, under the light of modern results about Vector Addition System with States (VASS). We show that emptiness is decidable for 2-OCNs over bounded languages (i.e., languages contained in a₁^* a₂^* ⋯ a_k^*), and decidable and Ackermann-complete for sweeping 2-OCNs, where the head direction only changes at the end-markers. Both decidability results revolve around reducing the problem to VASS reachability, but they rely on strikingly different approaches. In a second part, we study the expressive power of 2-OCNs, showing an array of connections between bounded languages, sweeping 2-OCNs, and semilinear languages. Most noteworthy among these connections, is that the bounded languages recognized by sweeping 2-OCNs are precisely those that are semilinear. Finally, we establish an intricate pumping lemma for 2-DOCNs and use it to show that there are OCN languages that are not 2-DOCN recognizable, improving on the known result that there are such 2-OCN languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata extensions
Keywords
  • Counter Net
  • Two way
  • Automata

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References

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