How Many Cooks Spoil the Soup?

TL;DR

This paper investigates the maximum parallelism in population protocols, characterizing which predicates can be computed with high symmetry and establishing an impossibility result for the parity predicate's symmetry.

Contribution

It provides a partial characterization of predicates computable with maximum symmetry and proves a strong symmetry bound for the parity predicate in population protocols.

Findings

Characterization of predicates with maximum symmetry in population protocols

Impossibility result for computing parity with high symmetry

Bound on protocol symmetry for parity predicate

Abstract

In this work, we study the following basic question: "How much parallelism does a distributed task permit?" Our definition of parallelism (or symmetry) here is not in terms of speed, but in terms of identical roles that processes have at the same time in the execution. We initiate this study in population protocols, a very simple model that not only allows for a straightforward definition of what a role is, but also encloses the challenge of isolating the properties that are due to the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes. We (i) give a partial characterization of the set of predicates on input assignments that can be stably computed with maximum symmetry, i.e., $\Theta(N_{min})$, where $N_{min}$ is the minimum multiplicity of a state in the initial configuration, and (ii) we turn our attention to the remaining predicates and prove a strong impossibility result for the parity predicate: the inherent symmetry of any protocol that stably computes it is upper bounded by a constant that depends on the size of the protocol.