Abstract
Partial synchrony is a model of computation in many distributed algorithms and modern blockchains. Correctness of these algorithms requires the existence of bounds on message delays and on the relative speed of processes after reaching Global Stabilization Time (GST). This makes partially synchronous algorithms parametric in time bounds, which renders automated verification of partially synchronous algorithms challenging. In this paper, we present a case study on formal verification of both safety and liveness of a Chandra and Toueg failure detector that is based on partial synchrony. To this end, we specify the algorithm and the partial synchrony assumptions in three frameworks: \(\textsc {TLA}^+\), Ivy, and counter automata. Importantly, we tune our modeling to use the strength of each method: (1) We are using counters to encode message buffers with counter automata, (2) we are using first-order relations to encode message buffers in Ivy, and (3) we are using both approaches in \(\textsc {TLA}^+\). By running the tools for \(\textsc {TLA}^+\) (TLC and APALACHE) and counter automata (FAST), we demonstrate safety for fixed time bounds. This helped us to find the inductive invariants for fixed parameters, which we used as a starting point for the proofs with Ivy. By running Ivy, we prove safety for arbitrary time bounds. Moreover, we show how to verify liveness of the failure detector by reducing the verification problem to safety verification. Thus, both properties are verified by developing inductive invariants with Ivy. We conjecture that correctness of other partially synchronous algorithms may be proven by following the presented methodology.
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References
Aguilera, M.K., Delporte-Gallet, C., Fauconnier, H., Toueg, S.: On implementing omega in systems with weak reliability and synchrony assumptions. Distrib. Comput. 21(4), 285–314 (2008)
Aguilera, M.K., Delporte-Gallet, C., Fauconnier, H., Toueg, S.: Consensus with Byzantine failures and little system synchrony. In: International Conference on Dependable Systems and Networks (DSN), pp. 147–155. IEEE (2006)
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
André, É., Fribourg, L., Kühne, U., Soulat, R.: IMITATOR 2.5: a tool for analyzing robustness in scheduling problems. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 33–36. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32759-9_6
Atif, M., Mousavi, M.R., Osaiweran, A.: Formal verification of unreliable failure detectors in partially synchronous systems. In: Proceedings of the 27th ACM Symposium on Applied Computing (SAC), pp. 478–485 (2012). https://doi.org/10.1145/2245276.2245369
Bardin, S., Leroux, J., Point, G.: FAST extended release. In: Ball, T., Jones, R. (eds.) CAV 2006. LNCS, vol. 4144, pp. 63–66. Springer, Heidelberg (2006). https://doi.org/10.1007/11817963_9
Bloem, R., et al.: Decidability of Computing Theory. Morgan & Claypool Publishers (2015). https://doi.org/10.2200/S00658ED1V01Y201508DCT013
Bravo, M., Chockler, G., Gotsman, A.: Making Byzantine consensus live. In: DISC. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2020)
Buchman, E., Kwon, J., Milosevic, Z.: The latest gossip on BFT consensus. arXiv preprint arXiv:1807.04938 (2018)
Bunte, O.: The mCRL2 toolset for analysing concurrent systems. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11428, pp. 21–39. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17465-1_2
Chandra, T.D., Toueg, S.: Unreliable failure detectors for reliable distributed systems. J. ACM 43(2), 225–267 (1996)
Chaudhuri, K., Doligez, D., Lamport, L., Merz, S.: The TLA\(^{+}\) proof system: building a heterogeneous verification platform. In: Cavalcanti, A., Deharbe, D., Gaudel, M.-C., Woodcock, J. (eds.) ICTAC 2010. LNCS, vol. 6255, p. 44. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14808-8_3
Cousineau, D., Doligez, D., Lamport, L., Merz, S., Ricketts, D., Vanzetto, H.: TLA\(^{+}\) proofs. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 147–154. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32759-9_14
Drăgoi, C., Widder, J., Zufferey, D.: Programming at the edge of synchrony. In: Proceedings of the ACM on Programming Languages 4 (OOPSLA), pp. 1–30 (2020)
Dwork, C., Lynch, N., Stockmeyer, L.: Consensus in the presence of partial synchrony. J. ACM 35(2), 288–323 (1988)
Emerson, E.A., Namjoshi, K.S.: Reasoning about rings. In: POPL, pp. 85–94 (1995)
Galois, I.: Ivy proofs of tendermint. https://github.com/tendermint/spec/tree/master/ivy-proofs. Accessed December 2020
Konnov, I., Kukovec, J., Tran, T.H.: TLA\(^{+}\) model checking made symbolic. In: Proceedings of the ACM on Programming Languages 3 (OOPSLA), pp. 1–30 (2019)
Konnov, I., Lazić, M., Veith, H., Widder, J.: Para\(^2\): parameterized path reduction, acceleration, and SMT for reachability in threshold-guarded distributed algorithms. Formal Methods Syst. Design 51(2), 270–307 (2017)
Konnov, I., Lazić, M., Veith, H., Widder, J.: A short counterexample property for safety and liveness verification of fault-tolerant distributed algorithms. In: POPL, pp. 719–734 (2017)
Kuppe, M.A., Lamport, L., Ricketts, D.: The TLA\(^{+}\) toolbox. arXiv preprint arXiv:1912.10633 (2019)
Lamport, L.: Specifying Systems: The TLA\(^{+}\) Language and Tools for Hardware and Software Engineers. Addison-Wesley, Boston (2002)
Lamport, L.: Using TLC to check inductive invariance (2018)
Larrea, M., Arevalo, S., Fernndez, A.: Efficient algorithms to implement unreliable failure detectors in partially synchronous systems. In: Jayanti, P. (ed.) DISC 1999. LNCS, vol. 1693, pp. 34–49. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48169-9_3
Larsen, K.G., Pettersson, P., Yi, W.: UPPAAL in a nutshell. Int. J. Softw. Tools Technol. Transfer 1(1–2), 134–152 (1997)
Lime, D., Roux, O.H., Seidner, C., Traonouez, L.-M.: Romeo: a parametric model-checker for petri nets with stopwatches. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 54–57. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00768-2_6
Lynch, N.A., Tuttle, M.R.: An Introduction to Input/Output Automata. Laboratory for Computer Science, Massachusetts Institute of Technology (1988)
McMillan, K.L.: Ivy. https://microsoft.github.io/ivy/. Accessed December 2020
McMillan, K.L., Padon, O.: Ivy: a multi-modal verification tool for distributed algorithms. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12225, pp. 190–202. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53291-8_12
Roscoe, A.W.: Understanding Concurrent Systems. Springer, Cham (2010)
Stoilkovska, I., Konnov, I., Widder, J., Zuleger, F.: Verifying safety of synchronous fault-tolerant algorithms by bounded model checking. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11428, pp. 357–374. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17465-1_20
Tran, T.H., Konnov, I., Widder, J.: FORTE2021-FD. https://github.com/banhday/forte2021-fd. Accessed April 2021
Tran, T.H., Konnov, I., Widder, J.: Specifications of the Chandra and Toueg failure detector in TLA\(^{+}\), and Ivy. https://zenodo.org/record/4687714#.YHcBeBKxVH4. Accessed April 2021
Tran, T.-H., Konnov, I., Widder, J.: Cutoffs for symmetric point-to-point distributed algorithms. In: Georgiou, C., Majumdar, R. (eds.) NETYS 2020. LNCS, vol. 12129, pp. 329–346. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-67087-0_21
Yin, M., Malkhi, D., Reiter, M.K., Gueta, G.G., Abraham, I.: Hotstuff: BFT consensus with linearity and responsiveness. In: PODC, pp. 347–356 (2019)
Yu, Y., Manolios, P., Lamport, L.: Model checking TLA\(^+\) specifications. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 54–66. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48153-2_6
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Supported by Interchain Foundation (Switzerland) and the Austrian Science Fund (FWF) via the Doctoral College LogiCS W1255.
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Tran, TH., Konnov, I., Widder, J. (2021). A Case Study on Parametric Verification of Failure Detectors. In: Peters, K., Willemse, T.A.C. (eds) Formal Techniques for Distributed Objects, Components, and Systems. FORTE 2021. Lecture Notes in Computer Science(), vol 12719. Springer, Cham. https://doi.org/10.1007/978-3-030-78089-0_8
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