Irreducibility and additivity of set agreement-oriented failure detector classes

Solving agreement problems (such as consensus and <i>k</i>-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 ◊ <i>S</i>_x and ◊ φ^y, 0≤ <i>x</i>, <i>y</i> ≤ <i>n</i>), shows that they can be "added" to provide a failure detector of the class Ω^z (a generalization of Ω). It also characterizes the power of such an "addition", namely, ◊ <b><i>S</i></b>_x + ◊ φ^y ➝ Ω^z ⇔ <i>x</i>+<i>y</i>+<i>z</i>><i>t</i>+1, where <i>t</i> is the maximum number of processes that can crash in a run. As an example, the paper shows that, while ◊ <b><i>S</i></b>_t allows solving 2-set agreement (and not consensus) and ◊ φ¹ allows solving <i>t</i>-set agreement (but not (<i>t</i>-1)-set agreement), their "addition" allows solving consensus. More generally, the paper studies the failure detector classes ◊ <b><i>S</i></b>_x, ◊ φ^y and Ω^z, and shows which reductions among these classes are possible and which are not. The paper presents also an Ω^k-based <i>k</i>-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 <i>k</i>-set agreement problem.