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<p><b>Abstract</b>—In this paper, we investigate the <it>k</it>-set consensus problem in asynchronous distributed systems. In this problem, each participating process begins the protocol with an input value and by the end of the protocol must decide on one value so that at most <it>k</it> total values are decided by all correct processes. We extend previous work by exploring several variations of the problem definition and model, including for the first time investigation of Byzantine failures. We show that the precise definition of the validity requirement, which characterizes what decision values are allowed as a function of the input values and whether failures occur, is crucial to the solvability of the problem. For example, we show that allowing default decisions in case of failures makes the problem solvable for most values of <it>k</it> despite a minority of failures, even in face of the most severe type of failures (Byzantine). We introduce six validity conditions for this problem (all considered in various contexts in the literature), and demarcate the line between possible and impossible for each case. In many cases, this line is different from the one of the originally studied <it>k</it>-set consensus problem.</p>
Agreement problems, Byzantine failures, consensus, crash failures, distributed systems, validity conditions.

R. De Prisco, D. Malkhi and M. Reiter, "On k-Set Consensus Problems in Asynchronous Systems," in IEEE Transactions on Parallel & Distributed Systems, vol. 12, no. , pp. 7-21, 2001.
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