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Counter Machines with Infrequent Reversals

Authors Alain Finkel , Shankara Narayanan Krishna , Khushraj Madnani , Rupak Majumdar , Georg Zetzsche

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Author Details

Alain Finkel
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, Gif-sur-Yvette, France
Shankara Narayanan Krishna
  • IIT Bombay, India
Khushraj Madnani
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Rupak Majumdar
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

Funding

Alain Finkel was funded by the Institut Universitaire de France and the Agence Nationale de la Recherche, grant BraVAS (ANR-17-CE40-0028). Khushraj Madnani and Rupak Majumdar were supported in part by the Deutsche Forschungsgemeinschaft project 389792660 TRR 248-CPEC.

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Alain Finkel, Shankara Narayanan Krishna, Khushraj Madnani, Rupak Majumdar, and Georg Zetzsche. Counter Machines with Infrequent Reversals. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.42

Abstract

Bounding the number of reversals in a counter machine is one of the most prominent restrictions to achieve decidability of the reachability problem. Given this success, we explore whether this notion can be relaxed while retaining decidability. 
To this end, we introduce the notion of an f-reversal-bounded counter machine for a monotone function f: ℕ → ℕ. In such a machine, every run of length n makes at most f(n) reversals. Our first main result is a dichotomy theorem: We show that for every monotone function f, one of the following holds: Either (i) f grows so slowly that every f-reversal bounded counter machine is already k-reversal bounded for some constant k or (ii) f belongs to Ω(log(n)) and reachability in f-reversal bounded counter machines is undecidable. This shows that classical reversal bounding already captures the decidable cases of f-reversal bounding for any monotone function f. The key technical ingredient is an analysis of the growth of small solutions of iterated compositions of Presburger-definable constraints. In our second contribution, we investigate whether imposing f-reversal boundedness improves the complexity of the reachability problem in vector addition systems with states (VASS). Here, we obtain an analogous dichotomy: We show that either (i) f grows so slowly that every f-reversal-bounded VASS is already k-reversal-bounded for some constant k or (ii) f belongs to Ω(n) and the reachability problem for f-reversal-bounded VASS remains Ackermann-complete. This result is proven using run amalgamation in VASS.
Overall, our results imply that classical restriction of reversal boundedness is a robust one.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Counter machines
  • reversal-bounded
  • reachability
  • decidability
  • complexity

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