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Congruence Relations for Büchi Automata

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Formal Methods (FM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 13047))

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Abstract

We revisit congruence relations for Büchi automata, which play a central role in automata-based formal verification. The size of the classical congruence relation is in \(3^{\mathcal {O}(n^{2})}\), where n is the number of states of the given Büchi automaton. We present improved congruence relations that can be exponentially coarser than the classical one. We further give asymptotically optimal congruence relations of size \(2^{\mathcal {O}(n \log n)}\). Based on these optimal congruence relations, we obtain an optimal translation from a Büchi automaton to a family of deterministic finite automata (FDFA), which can be made to accept either the original language or its complement. To the best of our knowledge, our construction is the first direct and optimal translation from Büchi automata to FDFAs.

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Notes

  1. 1.

    We use the normalized decomposition of UP-words defined in [17], which is different from the one given in [3]. Ours is a definition for a UP-word, while their definition is applied to a decomposition. However, this difference does not affect the definition of a saturated FDFA to be given later.

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Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions and comments about this paper. Work supported in part by the Guangdong Science and Technology Department (Grant No. 2018B010107004), NSF grants IIS-1527668, CCF-1704883 and IIS-1830549, and an award from the Maryland Procurement Office.

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Yong Li, Andrea Turrini & Lijun Zhang

  2. National Taiwan University, Taipei, Taiwan

    Yih-Kuen Tsay

  3. Institute of Intelligent Software, Guangzhou, China

    Andrea Turrini & Lijun Zhang

  4. Rice University, Houston, USA

    Moshe Y. Vardi

  5. University of Chinese Academy of Sciences, Beijing, China

    Lijun Zhang

Authors
  1. Yong Li
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  2. Yih-Kuen Tsay
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  3. Andrea Turrini
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  4. Moshe Y. Vardi
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  5. Lijun Zhang
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Correspondence to Lijun Zhang .

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Editors and Affiliations

  1. University of Twente, Enschede, The Netherlands

    Marieke Huisman

  2. NASA Ames Research Center, Moffett Field, CA, USA

    Corina Păsăreanu

  3. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Li, Y., Tsay, YK., Turrini, A., Vardi, M.Y., Zhang, L. (2021). Congruence Relations for Büchi Automata. In: Huisman, M., Păsăreanu, C., Zhan, N. (eds) Formal Methods. FM 2021. Lecture Notes in Computer Science(), vol 13047. Springer, Cham. https://doi.org/10.1007/978-3-030-90870-6_25

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  • DOI: https://doi.org/10.1007/978-3-030-90870-6_25

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