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Parikh Automata over Infinite Words

Authors Shibashis Guha , Ismaël Jecker , Karoliina Lehtinen , Martin Zimmermann

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Author Details

Shibashis Guha
  • Tata Institute of Fundamental Research, Mumbai, India
Ismaël Jecker
  • University of Warsaw, Poland
Karoliina Lehtinen
  • CNRS, Aix-Marseille University, LIS, Marseille, France
Martin Zimmermann
  • Aalborg University, Denmark


We want to thank an anonymous reviewer for proposing Lemma 14.

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Shibashis Guha, Ismaël Jecker, Karoliina Lehtinen, and Martin Zimmermann. Parikh Automata over Infinite Words. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 40:1-40:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the nondeterministic case; a result that sharply contrasts with the well-known hierarchy in the ω-regular setting. Furthermore, emptiness is shown decidable for Parikh automata with reachability or Büchi acceptance, but undecidable for safety and co-Büchi acceptance. Most importantly, we show decidability of model checking with specifications given by deterministic Parikh automata with safety or co-Büchi acceptance, but also undecidability for all other types of automata. Finally, solving games is undecidable for all types.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Parikh automata
  • ω-automata
  • Infinite Games


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