Sporadic solutions to zero-one exclusion tasks

Abstract

Zero-one exclusion is a family of distributed tasks indexed by n-bit Boolean signatures b[0],...b[n - 1]. We are interested in asynchronous computations where at most n + 1 asynchronous processes participate. They communicate with one another by reading and writing a shared memory, and halt after choosing a Boolean value. If m < n + 1 processes participate, then they must not all choose the value b[m - 1]. If all n + 1 processes participate, then they must not all choose the same value. It is easy to show that some instances of zero-one exclusion are computationally difficult, in the sense that they cannot be solved by any algorithm in which asynchronous processes communicate by reading and writing a shared memory. Can we characterize the Boolean signatures for which zero-one exclusion does have an asynchronous read-write algorithm? We give a partial answer, which we feel is interesting because of the way it ties together distributed computability, combinatorial topology, and elementary number theory. © 2014 Springer-Verlag.