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Counter abstraction is a powerful tool for parameterized model checking, if the number of local states of the concurrent processes is relatively small. In recent work, we introduced parametric interval counter abstraction that allowed us to verify the safety and liveness of threshold-based fault-tolerant distributed algorithms (FTDA). Due to state space explosion, applying this technique to distributed algorithms with hundreds of local states is challenging for state-of-the-art model checkers. In this paper, we demonstrate that reachability properties of FTDAs can be verified by bounded model checking. To ensure completeness, we need an upper bound on the diameter, i.e., on the longest distance between states. We show that the diameters of accelerated counter systems of FTDAs, and of their counter abstractions, have a quadratic upper bound in the number of local transitions. Our experiments show that the resulting bounds are sufficiently small to use bounded model checking for parameterized verification of reachability properties of several FTDAs, some of which have not been automatically verified before.

Automatic verification of threshold-based fault-tolerant distributed algorithms (FTDA) is challenging: they have multiple parameters that are restricted by arithmetic conditions, the number of processes and faults is parameterized, and the algorithm code is parameterized due to conditions counting the number of received messages. Recently, we introduced a technique that first applies data and counter abstraction and then runs bounded model checking (BMC). Given an FTDA, our technique computes an upper bound on the diameter of the system. This makes BMC complete: it always finds a counterexample, if there is an actual error. To verify state-of-the-art FTDAs, further improvement is needed. In this paper, we encode bounded executions over integer counters in SMT. We introduce a new form of offline partial order reduction that exploits acceleration and the structure of the FTDAs. This aggressively prunes the execution space to be explored by the solver. In this way, we verified safety of seven FTDAs that were out of reach before.

Distributed algorithms have many mission-critical applications ranging from embedded systems and replicated databases to cloud computing. Due to asynchronous communication, process faults, or network failures, these algorithms are difficult to design and verify. Many algorithms achieve fault tolerance by using threshold guards that, for instance, ensure that a process waits until it has received an acknowledgment from a majority of its peers. Consequently, domain-specific languages for fault-tolerant distributed systems offer language support for threshold guards.We introduce an automated method for model checking of safety and liveness of threshold-guarded distributed algorithms in systems where the number of processes and the fraction of faulty processes are parameters. Our method is based on a short counterexample property: if a distributed algorithm violates a temporal specification (in a fragment of LTL), then there is a counterexample whose length is bounded and independent of the parameters. We prove this property by (i) characterizing executions depending on the structure of the temporal formula, and (ii) using commutativity of transitions to accelerate and shorten executions. We extended the ByMC toolset (Byzantine Model Checker) with our technique, and verified liveness and safety of 10 prominent fault-tolerant distributed algorithms, most of which were out of reach for existing techniques.

Randomized fault-tolerant distributed algorithms pose a number of challenges for automated verification: (i) parameterization in the number of processes and faults, (ii) randomized choices and probabilistic properties, and (iii) an unbounded number of asynchronous rounds. This combination makes verification hard. Challenge (i) was recently addressed in the framework of threshold automata. We extend threshold automata to model randomized consensus algorithms that perform an unbounded number of asynchronous rounds. For non-probabilistic properties, we show that it is necessary and sufficient to verify these properties under round-rigid schedules, that is, schedules where processes enter round r only after all processes finished round $$r-1$$ r - 1 . For almost-sure termination, we analyze these algorithms under round-rigid adversaries, that is, fair adversaries that only generate round-rigid schedules. This allows us to do compositional and inductive reasoning that reduces verification of the asynchronous multi-round algorithms to model checking of a one-round threshold automaton. We apply this framework and automatically verify the following classic algorithms: Ben-Or’s and Bracha’s seminal consensus algorithms for crashes and Byzantine faults, 2-set agreement for crash faults, and RS-Bosco for the Byzantine case.

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Abstract. Fault-tolerant distributed algorithms are central for building reliable, spatially distributed systems. In order to ensure that these algorithms actually make systems more reliable, we must ensure that these algorithms are actually correct. Unfortunately, model checking state-ofthe-art fault-tolerant distributed algorithms (such as Paxos) is currently out of reach except for very small systems. In order to be eventually able to automatically verify such fault-tolerant distributed algorithms also in larger systems, several problems have to be addressed. In this paper, we consider modeling and verification of fault-tolerant algorithms that basically only contain threshold guards to control the flow of the algorithm. As threshold guards are widely used in fault-tolerant distributed algorithms (and also in Paxos) efficient methods to handle them bring us closer to the above mentioned goal. As case study we use the reliable broadcasting algorithm by Srikanth and Toueg that tolerates even Byzantine faults. We show how one can model this basic fault-tolerant distributed algorithm in Promela such that safety and liveness properties can be efficiently verified in Spin. We provide experimental data also for other distributed algorithms.