More Choices Allow More Faults: Set Consensus Problems in Totally Asynchronous Systems

Abstract

We define the k-SET CONSENSUS PROBLEM as an extension of the CONSENSUS problem, where each processor decides on a single value such that the set of decided values in any run is of size at most k. We require the agreement condition that all values decided upon are initial values of some processor. We show that the problem has a simple (k−1)-resilient protocol in a totally asynchronous system. In an attempt to come up with a matching lower bound on the number of failures, we study the uncertainty condition, which requires that there must be some initial configuration from which all possible input values can be decided. We prove using a combinatorial argument that any k-resilient protocol for the k-set agreement problem would satisfy the uncertainty condition, while this is not true for any (k−1)-resilient protocol. This result seems to strengthen the conjecture that there is no k-resilient protocol for this problem. We prove this result for a restricted class of protocols. Our motivation for studying this problem is to test whether the number of choices allowed to the processors is related to the number of faults. We hope that this will provide intuition towards achieving better bounds for more practical problems that arise in distributed computing, e.g., the renaming problem. The larger goal is to characterize the boundary between possibility and impossibility in asynchronous systems given multiple faults.