The 0–1-Exclusion Families of Tasks

Abstract

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 collection of families of tasks called 0-1 Exclusion tasks, and show that families in this collection are interesting.

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. A family of tasks, one for each n, is created by considering an infinite sequence of bits b(k), k = 2,3,..., such that the sequence that specifies instant n, is a prefix of the sequence that specifies the n + 1’st instance.

Only one family in the collection, the one specified by b(1) = b(2) = ...= 1, was implicitly considered in the past and shown to be equivalent to Set- Consensus. In this initial investigation of the whole collection we show that not all of its members are created equal. We take the family specified by b(1) = 1,b(2) = b(3) = ... = 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 collection. It is sandwiched between Set-Consensus 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 families, and make few conjectures about the implementations relationships among members of the collection, as well as possibly tasks outside it.