## Abstract

We address the problem of solving a task T = (T_{1}, ⋯T _{m}) (called (m, 1)-BG), in which a processor returns in an arbitrary one of m simultaneous consensus subtasks T_{1}, ⋯T_{m}. Processor p_{i} submits to T an input vector of proposals (prop _{i,1}, ⋯, prop_{i, m}), one entry per subtask, and outputs, from just one subtask ℓ, a pair (ℓ, prop_{j, l}) for some j. All processors that output at ℓ 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 T_{j} we allow k-set consensus rather than consensus, i.e., for each ℓ, the outputs satisfy |{j|prop _{j,ℓ}}| ≤ k, then the same algorithm solves T if m ≥ ⌊d/k⌋. What is the power of T = (T_{1},⋯, T _{m}) 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 p_{i} can submit the vector (id_{i}, id _{i}, ⋯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 T_{j} is just a binary-consensus rather than consensus? Then we get the novel problem that was recently introduced of the Committee-Decision. 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 sub-tasks, 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 binary-set-consensus is well defined. We have then showed that binary-set-consensus is equivalent to set consensus as it was with consensus.

Original language | English |
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Title of host publication | Distributed Computing and Networking - 8th International Conference, ICDCN 2006, Proceedings |

Pages | 331-341 |

Number of pages | 11 |

DOIs | |

State | Published - 2006 |

Event | 8th International Conference on Distributed Computing and Networking, ICDCN 2006 - Guwahati, India Duration: 27 Dec 2006 → 30 Dec 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4308 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 8th International Conference on Distributed Computing and Networking, ICDCN 2006 |
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Country/Territory | India |

City | Guwahati |

Period | 27/12/06 → 30/12/06 |