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Generic Emptiness Check for Fun and Profit

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Automated Technology for Verification and Analysis (ATVA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11781))

Abstract

We present a new algorithm for checking the emptiness of \(\omega \)-automata with an Emerson-Lei acceptance condition (i.e., a positive Boolean formula over sets of states or transitions that must be visited infinitely or finitely often). The algorithm can also solve the model checking problem of probabilistic positiveness of MDP under a property given as a deterministic Emerson-Lei automaton. Although both these problems are known to be NP-complete and our algorithm is exponential in general, it runs in polynomial time for simpler acceptance conditions like generalized Rabin, Streett, or parity. In fact, the algorithm provides a unifying view on emptiness checks for these simpler automata classes. We have implemented the algorithm in Spot and PRISM and our experiments show improved performance over previous solutions.

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Notes

  1. 1.

    As this problem can be understood by the average Sudoku player, more instances can be found at https://adl.github.io/genem-exp/examples/ either to practice the algorithm by hand, or as an entertaining prophylaxis of Alzheimer’s disease.

  2. 2.

    The original definition was state-based, which means that marks were on states and not transitions. As \(\omega \)-automata with state-based acceptance can be easily converted to transition-based acceptance without changing the transition structure, we focus on transition-based \(\omega \)-automata only.

  3. 3.

    In fact, Fig. 1 comes from a product of a Rabin and a Streett automaton.

  4. 4.

    See https://adl.github.io/genem-exp/bench-app1/ to reproduce.

  5. 5.

    See https://adl.github.io/genem-exp/bench-app2/ to reproduce.

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Acknowledgement

This research was partially supported by the DFG through the DFG-project BA-1679/11-1, the DFG-project BA-1679/12-1, the Collaborative Research Centers CRC 912 (HAEC) and CRC 248 (DFG grant 389792660 as part of TRR 248), the Cluster of Excellence EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), the Research Training Groups QuantLA (GRK 1763), by F.R.S.-FNRS through the grant F.4520.18 (ManySynth), and by the Czech Science Foundation through the grant GA19-24397S.

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Authors and Affiliations

  1. Technische Universität Dresden, Dresden, Germany

    Christel Baier, Joachim Klein & David Müller

  2. University of Mons, Mons, Belgium

    František Blahoudek

  3. LRDE, EPITA, Le Kremlin-Bicêtre, France

    Alexandre Duret-Lutz

  4. Masaryk University, Brno, Czech Republic

    Jan Strejček

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  1. Christel Baier
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Correspondence to Alexandre Duret-Lutz .

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Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    Yu-Fang Chen

  2. DENSO AUTOMOTIVE Deutschland GmbH, Eching, Germany

    Chih-Hong Cheng

  3. Institute of Computer Science, TU München, Munich, Germany

    Javier Esparza

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Baier, C., Blahoudek, F., Duret-Lutz, A., Klein, J., Müller, D., Strejček, J. (2019). Generic Emptiness Check for Fun and Profit. In: Chen, YF., Cheng, CH., Esparza, J. (eds) Automated Technology for Verification and Analysis. ATVA 2019. Lecture Notes in Computer Science(), vol 11781. Springer, Cham. https://doi.org/10.1007/978-3-030-31784-3_26

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