Irreducibility and additivity of set agreement-oriented failure detector classes

• Authors: Achour Mostefaoui;Sergio Rajsbaum;Michel Raynal;Corentin Travers
• Venue: Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
• Year: 2006

Solving agreement problems (such as consensus and k-set agreement) in asynchronous distributed systems prone to process failures has been shown to be impossible. To circumvent this impossibility, distributed oracles (also called unreliable failure detectors) have been introduced. A failure detector provides information on failures, and a failure detector class is defined by a set of abstract properties that encapsulate (and hide) synchrony assumptions. Some failure detector classes have been shown to be the weakest to solve some agreement problems (e.g., Ω is the weakest class of failure detectors that allow solving the consensus problem in asynchronous systems where a majority of processes do not crash).This paper considers several failure detector classes and focuses on their additivity or their irreducibility. It mainly investigates two families of failure detector classes (denoted ◊ Sx and ◊ φy, 0≤ x, y ≤ n), shows that they can be "added" to provide a failure detector of the class Ωz (a generalization of Ω). It also characterizes the power of such an "addition", namely, ◊ Sx + ◊ φy ➝ Ωz ⇔ x+y+zt+1, where t is the maximum number of processes that can crash in a run. As an example, the paper shows that, while ◊ St allows solving 2-set agreement (and not consensus) and ◊ φ1 allows solving t-set agreement (but not (t-1)-set agreement), their "addition" allows solving consensus. More generally, the paper studies the failure detector classes ◊ Sx, ◊ φy and Ωz, and shows which reductions among these classes are possible and which are not. The paper presents also an Ωk-based k-set agreement protocol. In that sense, it can be seen as a step toward the characterization of the weakest failure detector that allows solving the k-set agreement problem.