Verification of the randomized consensus algorithm of Aspnes and Herlihy: a case study

The Probabilistic I/O Automaton model of [Seg95] is used as the basis for a formal presentation and proof of the randomized consensus algorithm of Aspnes and Herlihy. The algorithm guarantees termination within expected polynomial time. The Aspnes-Herlihy algorithm is a rather complex algorithm. Processes move through a succession of asynchronous rounds, attempting to agree at each round. At each round, the agreement attempt involves a distributed random walk. The algorithm is hard to analyze because of its use of nontrivial results of probability theory (specifically, random walk theory), because of its complex setting, including asynchrony and both nondeterministic and probabilistic choice, and because of the interplay among several different sub-protocols. We formalize the Aspnes-Herlihy algorithm using probabilistic I/O automata. In doing so, we decompose it formally into three subprotocols: one to carry out the agreement attempts, one to conduct the random walks, and one to implement a shared counter needed by the random walks. Properties of all three subprotocols are proved separately, and combined using general results about automaton composition. It turns out that most of the work involves proving non-probabilistic properties (invariants, simulation mappings, non-probabilistic progress properties, etc.). The probabilistic reasoning is isolated to a few small sections of the proof. The task of carrying out this proof has led us to develop several general proof techniques for probabilistic I/O automata. These include ways to combine expectations for different complexity measures, to compose expected complexity properties, to convert probabilistic claims to deterministic claims, to use abstraction mappings to prove probabilistic properties, and to apply random walk theory in a distributed computational setting. We apply all of these techniques to analyze the expected complexity of the algorithm.