Revisiting Parameter Synthesis for One-Counter Automata

Authors Guillermo A. Pérez , Ritam Raha

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Guillermo A. Pérez
  • University of Antwerp, Flanders Make, Belgium
Ritam Raha
  • University of Antwerp, Belgium
  • LaBRI, University of Bordeaux, France


We thank all anonymous reviewers who carefully read earlier version of this work and helped us polish the paper. We also thank Michaël Cadilhac, Nathanaël Fijalkow, Philip Offtermatt, and Mikhail R. Starchak for useful feedback; Antonia Lechner and James Worrell for having brought the inconsistencies in [Marius Bozga and Radu Iosif, 2005; Antonia Lechner, 2015] to our attention; Radu Iosif and Marius Bozga for having pointed out precisely what part of the argument in both papers is incorrect.

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Guillermo A. Pérez and Ritam Raha. Revisiting Parameter Synthesis for One-Counter Automata. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study the synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some ω-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called ∀∃_RPAD^+, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of ∀∃_RPAD^+, (ii) we prove that the synthesis problem is decidable in general and in 2NEXP for several fixed ω-regular properties. Finally, (iii) we give polynomial-space algorithms for the special cases of the problem where parameters can only be used in counter tests.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantitative automata
  • Theory of computation → Logic and verification
  • Parametric one-counter automata
  • Reachability
  • Software Verification


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