Distributed protocols such as Paxos play an important role in many computer systems. Therefore, a bug in a distributed protocol may have tremendous effects. Accordingly, a lot of effort has been invested in verifying such protocols. However, checking invariants of such protocols is undecidable and hard in practice, as it requires reasoning about an unbounded number of nodes and messages. Moreover, protocol actions and invariants involve both quantifier alternations and higher-order concepts such as set cardinalities and arithmetic.This paper makes a step towards automatic verification of such protocols. We aim at a technique that can verify correct protocols and identify bugs in incorrect protocols. To this end, we develop a methodology for deductive verification based on effectively propositional logic (EPR)-a decidable fragment of first-order logic (also known as the Bernays-Schönfinkel-Ramsey class). In addition to decidability, EPR also enjoys the finite model property, allowing to display violations as finite structures which are intuitive for users. Our methodology involves modeling protocols using general (uninterpreted) first-order logic, and then systematically transforming the model to obtain a model and an inductive invariant that are decidable to check. The steps of the transformations are also mechanically checked, ensuring the soundness of the method. We have used our methodology to verify the safety of Paxos, and several of its variants, including Multi-Paxos, Vertical Paxos, Fast Paxos, Flexible Paxos and Stoppable Paxos. To the best of our knowledge, this work is the first to verify these protocols using a decidable logic, and the first formal verification of Vertical Paxos, Fast Paxos and Stoppable Paxos.

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We develop a new technique for verifying temporal properties of infinite-state (distributed) systems. The main idea is to reduce the temporal verification problem to the problem of verifying the safety of infinite-state systems expressed in first-order logic. This allows to leverage existing techniques for safety verification to verify temporal properties of interesting distributed protocols, including some that have not been mechanically verified before. We model infinite-state systems using first-order logic, and use first-order temporal logic (FO-LTL) to specify temporal properties. This general formalism allows to naturally model distributed systems, while supporting both unbounded-parallelism (where the system is allowed to dynamically create processes), and infinite-state per process. The traditional approach for verifying temporal properties of infinite-state systems employs well-founded relations (e.g. using linear arithmetic ranking functions). In contrast, our approach is based the idea of fair cycle detection. In finite-state systems, temporal verification can always be reduced to fair cycle detection (a system contains a fair cycle if it revisits a state after satisfying all fairness constraints). However, with both infinitely many states and infinitely many fairness constraints, a straightforward reduction to fair cycle detection is unsound. To regain soundness, we augment the infinite-state transition system by a dynamically computed finite set, that exploits the locality of transitions. This set lets us define a form of fair cycle detection that is sound in the presence of both infinitely many states, and infinitely many fairness constraints. Our approach allows a new style of temporal verification that does not explicitly involve ranking functions. This fits well with pure first-order verification which does not explicitly reason about numerical values. In particular, it can be used with effectively propositional first-order logic (EPR), in which case checking verification conditions is decidable. We applied our technique to verify temporal properties of several interesting protocols. To the best of our knowledge, we have obtained the first mechanized liveness proof for both TLB Shootdown, and Stoppable Paxos. CCS Concepts: • Theory of computation → Logic and verification; Modal and temporal logics; • Software and its engineering → Formal methods;

Proof automation can substantially increase productivity in formal verification of complex systems. However, unpredictablility of automated provers in handling quantified formulas presents a major hurdle to usability of these tools. We propose to solve this problem not by improving the provers, but by using a modular proof methodology that allows us to produce decidable verification conditions. Decidability greatly improves predictability of proof automation, resulting in a more practical verification approach. We apply this methodology to develop verified implementations of distributed protocols, demonstrating its effectiveness. CCS Concepts • Software and its engineering → Formal software verification;

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