Minimising Good-For-Games Automata Is NP-Complete

Author Sven Schewe

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Sven Schewe
  • University of Liverpool, UK


Many thanks to Patrick Totzke and Karoliina Lehtinen for valuable feedback and pointers to beautiful related works, as well as the constructive feedback of the reviewers.

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Sven Schewe. Minimising Good-For-Games Automata Is NP-Complete. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 56:1-56:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


This paper discusses the hardness of finding minimal good-for-games (GFG) Büchi, Co-Büchi, and parity automata with state based acceptance. The problem appears to sit between finding small deterministic and finding small nondeterministic automata, where minimality is NP-complete and PSPACE-complete, respectively. However, recent work of Radi and Kupferman has shown that minimising Co-Büchi automata with transition based acceptance is tractable, which suggests that the complexity of minimising GFG automata might be cheaper than minimising deterministic automata. We show for the standard state based acceptance that the minimality of a GFG automaton is NP-complete for Büchi, Co-Büchi, and parity GFG automata. The proofs are a surprisingly straight forward generalisation of the proofs from deterministic Büchi automata: they use a similar reductions, and the same hard class of languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Good-for-Games Automata
  • Automata Minimisation


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