Distributed architectures are used to improve performance and reliability of various systems. Examples include drone swarms and load-balancing servers. An important capability of a distributed architecture is the ability to reach consensus among all its nodes. Several consensus algorithms have been proposed, and many of these algorithms come with intricate proofs of correctness, that are not mechanically checked. In the controls community, algorithms often achieve consensus asymptotically , e.g., for problems such as the design of human control systems, or the analysis of natural systems like bird flocking. This is in contrast to exact consensus algorithm such as Paxos, which have received much more recent attention in the formal methods community. This paper presents the first formal proof of an asymptotic consensus algorithm, and addresses various challenges in its formalization. Using the Coq proof assistant, we verify the correctness of a widely used consensus algorithm in the distributed controls community, the Weighted-Mean Subsequence Reduced (W-MSR) algorithm . We formalize the necessary and sufficient conditions required to achieve resilient asymptotic consensus under the assumed attacker model. During the formalization, we clarify several imprecisions in the paper proof, including an imprecision on quantifiers in the main theorem.

We present AVR, a push-button model checker for verifying state transition systems directly at the source-code level. AVR uses information embedded in the word-level syntax of the design representation to automatically perform scalable model checking by combining a novel syntax-guided abstraction-refinement technique with a word-level implementation of the IC3 algorithm. AVR provides independently-verifiable certificates that offer provable assurance and are easy to relate to the word-level system. Moreover, proof certificates can be further used in innovative ways to extract key design information and are useful in a growing number of applications.