The 0− 1-Exclusion Family of Tasks

Interesting tasks are scarce. Yet they are essential as an investigation material, if we are to understand the structure of the tasks world. We propose a new family of tasks called 0-1 Exclusion tasks, and show it is an interesting family. A 0-1 Exclusion task on n processors is specified by a sequence of n− 1 bits b(1), b(2), ..., b(n− 1). For participating set of size k, 0 < k < n, each processor is to output 0 or 1 but they should not all output b(k). When the participating set is of size n then they should all output neither all 0’s nor all 1’s. Only one member of this family, the one specified by b(1) = b(2) = ... = b(n − 1) = 1, was implicitly considered in the past and shown to be equivalent to Set-Consensus. In this initial investigation of the whole family we show that not all of its members are created equal. We take the member specified by b(1) = 1, b(2) = ... = b(n− 1) = 0, and show that it is read-write unsolvable for all n, but is strictly weaker than Set-Consensus for n odd. We show some general results about the whole family. The family is sandwiched between SetConsensus from above and Weak-Symmetry-Breaking from below. Any Black-Box of n ports that solves a 0-1 Exclusion task, can be used to solve that task for n processors with ids from unbounded domain. Finally we show an intriguing relation between Strong-Renaming and the 0-1 Exclusion family, and make few conjectures about the implementations relationships among members of of the family, as well as possibly tasks outside it.