Sub-Consensus hierarchy conjecture is false

Each query to failure detector ¬Ωk outputs n− k processes; at least one correct process is eventually never output. The “folklore” sub-Consensus hierarchy conjecture states states that any task not solvable with ¬Ωk requires ¬Ωk−1. This paper disproves this conjecture for symmetric, participation-aware tasks. Consider any sequence (ki) = k1, . . . , kn with ki ≤ ki+1 ≤ ki +1. The agreement task T(ki) decides on at most ki proposals, where i is the number of participating processes. Detector D(ki) consists of sub-detectors D1, . . . , Dn, such that at least one of D1, . . . , Dkc behaves like Ω, where c is the number of correct processes. This paper shows that detector D(ki) can implement task T(k′ i) iff ki ≤ k ′ i for all i. As a result, no two different D(ki)’s are equivalent. Moreover, D(ki) is the weakest failure detector for T(ki). The number of D(ki) detectors (2 n−1) exceeds the number of ¬Ωi detectors (n), violating the sub-Consensus conjecture.