A Classification of Degenerate Loop Agreement

Loop agreement is a type of distributed decision tasks including many well-known tasks such as set agreement, simplex agreement, and approximation agreement. Because of its elegant combinatorial structure and its important role in the decidability problem of distributed decision tasks, loop agreement has been thoroughly investigated. A classification of loop agreement tasks has been proposed, based on their relative computational power: tasks are in the same class if and only if they can implement each other. However, the classification does not cover such important tasks as consensus, because any loop agreement task allows up to three distinct output values in an execution. So, this paper considers classifying a variation of loop agreement, called degenerate loop agreement, which includes consensus. A degenerate loop agreement task is defined in terms of its decision space and two distinguished vertices in the space. It is shown that there are exactly two equivalence classes of degenerate loop agreement tasks: one represented by the trivial task, and the other by consensus. The classification is totally determined by connectivity of the decision space of a task; if the distinguished points are connected in the space, the task is equivalent to the trivial task, otherwise to consensus.