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The Complexity of Simplifying ω-Automata Through the Alternating Cycle Decomposition

Authors Antonio Casares , Corto Mascle

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Author Details

Antonio Casares
  • University of Warsaw, Poland
Corto Mascle
  • LaBRI, Université de Bordeaux, France

Funding

Antonio Casares: Supported by the Polish National Science Centre (NCN) grant "Polynomial finite state computation" (2022/46/A/ST6/00072).

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Antonio Casares and Corto Mascle. The Complexity of Simplifying ω-Automata Through the Alternating Cycle Decomposition. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.35

Abstract

In 2021, Casares, Colcombet and Fijalkow introduced the Alternating Cycle Decomposition (ACD), a structure used to define optimal transformations of Muller into parity automata and to obtain theoretical results about the possibility of relabelling automata with different acceptance conditions. In this work, we study the complexity of computing the ACD and its DAG-version, proving that this can be done in polynomial time for suitable representations of the acceptance condition of the Muller automaton. As corollaries, we obtain that we can decide typeness of Muller automata in polynomial time, as well as the parity index of the languages they recognise. Furthermore, we show that we can minimise in polynomial time the number of colours (resp. Rabin pairs) defining a Muller (resp. Rabin) acceptance condition, but that these problems become NP-complete when taking into account the structure of an automaton using such a condition.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Omega-regular languages
  • Muller automata
  • Zielonka tree

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