Skip to main content

Complexity of Verification and Synthesis of Threshold Automata

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 12302)

Abstract

Threshold automata are a formalism for modeling and analyzing fault-tolerant distributed algorithms, recently introduced by Konnov, Veith, and Widder, describing protocols executed by a fixed but arbitrary number of processes. We conduct the first systematic study of the complexity of verification and synthesis problems for threshold automata. We prove that the coverability, reachability, safety, and liveness problems are NP-complete, and that the bounded synthesis problem is \(\varSigma _p^2\) complete. A key to our results is a novel characterization of the reachability relation of a threshold automaton as an existential Presburger formula. The characterization also leads to novel verification and synthesis algorithms. We report on an implementation, and provide experimental results.

Keywords

  • Threshold automata
  • Distributed algorithms
  • Parameterized verification

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 787367 (PaVeS).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-59152-6_8
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-59152-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Notes

  1. 1.

    A full version of this paper containing additional details and proofs can be found at https://arxiv.org/abs/2007.06248.

References

  1. Bloem, R., et al.: Decidability of Parameterized Verification. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, San Rafael (2015)

    Google Scholar 

  2. Blondin, M., Haase, C., Mazowiecki, F.: Affine extensions of integer vector addition systems with states. In: CONCUR. LIPIcs, vol. 118, pp. 14:1–14:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  3. Bracha, G., Toueg, S.: Asynchronous consensus and broadcast protocols. J. ACM 32(4), 824–840 (1985)

    MathSciNet  CrossRef  Google Scholar 

  4. Brasileiro, F., Greve, F., Mostefaoui, A., Raynal, M.: Consensus in one communication step. In: Malyshkin, V. (ed.) PaCT 2001. LNCS, vol. 2127, pp. 42–50. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44743-1_4

    CrossRef  Google Scholar 

  5. Chandra, T.D., Toueg, S.: Unreliable failure detectors for reliable distributed systems. J. ACM 43(2), 225–267 (1996)

    MathSciNet  CrossRef  Google Scholar 

  6. Dobre, D., Suri, N.: One-step consensus with zero-degradation. In: DSN, pp. 137–146 (2006)

    Google Scholar 

  7. Dufourd, C., Finkel, A., Schnoebelen, P.: Reset nets between decidability and undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055044

    CrossRef  MATH  Google Scholar 

  8. Esparza, J.: Decidability and complexity of Petri net problems—An introduction. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1491, pp. 374–428. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-65306-6_20

    CrossRef  MATH  Google Scholar 

  9. Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: LICS, pp. 352–359. IEEE Computer Society (1999)

    Google Scholar 

  10. Esparza, J., Nielsen, M.: Decidability issues for petri nets - a survey. Bull. EATCS 52, 244–262 (1994)

    MATH  Google Scholar 

  11. German, S.M., Sistla, A.P.: Reasoning about systems with many processes. J. ACM 39(3), 675–735 (1992)

    MathSciNet  CrossRef  Google Scholar 

  12. Guerraoui, R.: Non-blocking atomic commit in asynchronous distributed systems with failure detectors. Distrib. Comput. 15(1), 17–25 (2002). https://doi.org/10.1007/s446-002-8027-4

    CrossRef  Google Scholar 

  13. Haase, C.: A survival guide to Presburger arithmetic. ACM SIGLOG News 5(3), 67–82 (2018)

    CrossRef  Google Scholar 

  14. Konnov, I., Lazic, M., Veith, H., Widder, J.: Para\({}^{\text{2 }}\): parameterized path reduction, acceleration, and SMT for reachability in threshold-guarded distributed algorithms. Formal Methods Syst. Des. 51(2), 270–307 (2017)

    CrossRef  Google Scholar 

  15. Konnov, I., Veith, H., Widder, J.: On the completeness of bounded model checking for threshold-based distributed algorithms: reachability. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 125–140. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44584-6_10

    CrossRef  Google Scholar 

  16. Konnov, I., Veith, H., Widder, J.: On the completeness of bounded model checking for threshold-based distributed algorithms: reachability. Inf. Comput. 252, 95–109 (2017)

    MathSciNet  CrossRef  Google Scholar 

  17. Konnov, I., Widder, J.: ByMC: Byzantine model checker. In: Margaria, T., Steffen, B. (eds.) ISoLA 2018. LNCS, vol. 11246, pp. 327–342. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03424-5_22

    CrossRef  Google Scholar 

  18. Konnov, I.V., Lazic, M., Veith, H., Widder, J.: A short counterexample property for safety and liveness verification of fault-tolerant distributed algorithms. In: POPL 2017, pp. 719–734 (2017)

    Google Scholar 

  19. Kukovec, J., Konnov, I., Widder, J.: Reachability in parameterized systems: all flavors of threshold automata. In: CONCUR, pp. 19:1–19:17 (2018)

    Google Scholar 

  20. Ladner, R.E.: The circuit value problem is log space complete for p. SIGACT News 7(1), 18–20 (1975)

    CrossRef  Google Scholar 

  21. Lazić, M., Konnov, I., Widder, J., Bloem, R.: Synthesis of distributed algorithms with parameterized threshold guards. In: OPODIS. LIPIcs, vol. 95, pp. 32:1–32:20 (2017)

    Google Scholar 

  22. Mostéfaoui, A., Mourgaya, E., Parvédy, P.R., Raynal, M.: Evaluating the condition-based approach to solve consensus. In: DSN, pp. 541–550 (2003)

    Google Scholar 

  23. Raynal, M.: A case study of agreement problems in distributed systems: non-blocking atomic commitment. In: HASE, pp. 209–214 (1997)

    Google Scholar 

  24. Schmitz, S., Schnoebelen, P.: The power of well-structured systems. CoRR abs/1402.2908 (2014)

    Google Scholar 

  25. Song, Y.J., van Renesse, R.: Bosco: one-step Byzantine asynchronous consensus. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 438–450. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87779-0_30

    CrossRef  Google Scholar 

  26. Srikanth, T., Toueg, S.: Simulating authenticated broadcasts to derive simple fault-tolerant algorithms. Distrib. Comput. 2, 80–94 (1987). https://doi.org/10.1007/BF01667080

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. R. Balasubramanian .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Balasubramanian, A.R., Esparza, J., Lazić, M. (2020). Complexity of Verification and Synthesis of Threshold Automata. In: Hung, D.V., Sokolsky, O. (eds) Automated Technology for Verification and Analysis. ATVA 2020. Lecture Notes in Computer Science(), vol 12302. Springer, Cham. https://doi.org/10.1007/978-3-030-59152-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-59152-6_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-59151-9

  • Online ISBN: 978-3-030-59152-6

  • eBook Packages: Computer ScienceComputer Science (R0)