What can be computed in an anonymous network, where nodes are not equipped with unique identifiers? It turns out that the answer to this question depends on the commitment of the nodes to their first computed output value: Two classes of problems solvable in anonymous networks are defined, where in the first class nodes are allowed to revoke their outputs and in the second class they are not. These two classes are then related to the class of all centrally solvable network problems, observing that the three classes form a strict linear hierarchy, and for several classic and/or characteristic problems in distributed computing, we determine the exact class to which they belong. Does this hierarchy exhibit complete problems? We answer this question in the affirmative by introducing the concept of a distributed oracle, thus establishing a more fine grained classification for distributed computability which we apply to the classic/characteristic problems. Among our findings is the observation that the three classes are characterized by the three pillars of distributed computing, namely, local symmetry breaking, coordination, and leader election.
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