Abstract
It is conjectured that the only way a failure detector (FD) can help solving n-process tasks is by providing k-set consensus for some \({k\in\{1,\ldots,n\}}\) among all the processes. It was recently shown by Zieliński that any FD that allows for solving a given n-process task that is unsolvable read-write wait-free, also solves (n − 1)-set consensus. In this paper, we provide a generalization of Zieliński’s result. We show that any FD that solves a colorless task that cannot be solved read-write k-resiliently, also solves k-set consensus. More generally, we show that every colorless task \({\mathcal{T}}\) can be characterized by its set consensus number: the largest \({k\in\{1,\ldots,n\}}\) such that \({\mathcal{T}}\) is solvable (k − 1)-resiliently. A task \({\mathcal{T}}\) with set consensus number k is, in the failure detector sense, equivalent to k-set consensus, i.e., a FD solves \({\mathcal{T}}\) if and only if it solves k-set consensus. As a corollary, we determine the weakest FD for solving k-set consensus in every environment, i.e., for all assumptions on when and where failures might occur.
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This paper combines results appeared in the Proceedings of the 28th ACM Symposium on Principles of Distributed Computing (PODC 2009) and the 23rd International Symposium on Distributed Computing (DISC 2009).
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Gafni, E., Kuznetsov, P. On set consensus numbers. Distrib. Comput. 24, 149–163 (2011). https://doi.org/10.1007/s00446-011-0142-8
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DOI: https://doi.org/10.1007/s00446-011-0142-8