(Set agreement). In each run, every process of the system pro- poses a value, and has to decide a value satisfying the following three properties:
(Set agreement). Set agreement captures a weaker form of consensus in which the agree- ment property is weakened as follows: – At most n — 1 different values are decided upon. — — That is, in any given run, the size of the set of the decision values is at most n 1. In particular, in runs in which the n processes propose n different values, instead of forcing the processes to agree on a single value, set agreement forces them to elimi- nate one of the proposed values. The set agreement problem as defined above is also called the (n 1)-set agreement problem [10]. While much weaker than consensus, as consensus, set agreement cannot be solved in crash-prone non-anonymous RW mem- ory systems [7, 16, 28] (and consequently cannot be solved in a crash-prone anony- mous memory systems either), but, as consensus, it can be solved when considering the weaker obstruction-freedom progress condition. ≥
(Set agreement). In each run, every process of the system proposes a value, and has to decide a value satisfying the following three properties:
(Set agreement). − − To illustrate some of the ideas that will be used in our message adversary for general k-set agreement, we start with the simple case of n 1-set agreement (also called set agreement ) first. Note that Theorem 7 does not apply here. To circumvent the impossibility result of Theorem 9, it suffices to strengthen the assumption of at most n 1 source components in every round such that the generation of too many decision values during the unstable period is ruled out. A straightforward way to achieve this is to just forbid n different decisions obtained in source components consisting of a single process. Achieving this is easy under the Σn−1-influence message adversary given in Definition 16, the name of which has been inspired by the Σn−1 failure detector [59]. j Definition 16 (Σn−1-influence message adversary). The message adversary Σn−1-MAJ is the set of all sequences of communication graphs (Gr)r>0 that satisfy the following: if each process pi ∈ Π becomes a single-node source component during a non-empty set of Intervals Xi, then any selection {I1, . . . , In} with Ii ∈ Xi for 1 ™ i ™ n, contains two distinct Ii = [a, b] and Ij = [aj, bj] such that sb ~ sat . 1210 It is easy to devise a set agreement algorithm that works correctly in a dynamic network under Defini- tion 16, provided (a bound on) n is known: In Algorithm 3, process pi maintains a proposal value vi, initially xi, and a decision value yi, initially ⊥, which are broadcast in every round. If pi receives no message from any other process in a round, it decides by setting yi = vi. If pi receives a message from some pj that has already decided (yj ⊥), it sets yi = yj. Otherwise, it updates vi to the maximum of vi and all received values vj. At the end of round n, a process that has not yet decided sets yi := vi, and all processes terminate. Algorithm 3 Set agreement algorithm for message adversary Σn−1-MAJ. Set agreement algorithm, code for process pi: ∈
