Simultaneous Consensus Tasks: A Tighter Characterization of Set-Consensus

We address the problem of solving a task T = (T1, ...Tm) (called (m, 1)-BG), in which a processor returns in an arbitrary one of m simultaneous consensus subtasks T1, ...Tm. Processor pi submits to T an input vector of proposals (propi,1, ..., propi,m), one entry per subtask, and outputs, from just one subtask , a pair ( , propj,l) for some j. All processors that output at output the same proposal. Let d be a bound on the number of distinct input vectors that may be submitted to T . For example, d = 3 if Democrats always vote Democrats across the board, and similarly for Republicans and Libertarians. A wait-free algorithm that immaterial of the number of processors solves T provided m ≥ d is presented. In addition, if in each Tj we allow k-set consensus rather than consensus, i.e., for each , the outputs satisfy |{j | propj, }| ≤ k, then the same algorithm solves T if m ≥ d/k . What is the power of T = (T1, ..., Tm) when given as a subroutine, to be used by any number of processors with any number of input vectors? Obviously, T solves m-set consensus since each processor pi can submit the vector (idi, idi, ...idi), but can m-set consensus solve T ? We show it does, and thus simultaneous consensus is a new characterization of set-consensus. Finally, what if each Tj is just a binary-consensus rather than consensus? Then we get the novel problem that was recently introduced of the CommitteeDecision. It was shown that for 3 processors and m = 2, the simultaneous binary-consensus is equivalent to (3, 2)-set consensus. Here, using a variation of our wait-free algorithms mentioned above, we show that a task, in which a processor is required to return in one of m simultaneous binary-consensus subtasks, when used by n processors, is equivalent to (n,m)-set consensus. Thus, while set-consensus unlike consensus, has no binary version, now that we characterize m-set consensus through simultaneous consensus, the notion of binaryset-consensus is well defined. We have then showed that binary-set-consensus is equivalent to set consensus as it was with consensus.