Large Flocks of Small Birds: On the Minimal Size of Population Protocols

TL;DR

This paper investigates the minimal memory size needed for population protocols to compute certain predicates, revealing that some can be computed with surprisingly small state complexity, including logarithmic and double-logarithmic bounds.

Contribution

It introduces bounds on the number of states required for population protocols to compute linear inequalities, including novel double-logarithmic state protocols and matching lower bounds.

Findings

Protocols for $x \,\geq\, n$ use $O(\log n)$ states.

Some predicates are computable with $O(\log\log n)$ states.

Matching lower bounds for 1-aware protocols are provided.

Abstract

Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates $x \geq n$ for $n \in \mathbb{N}$, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with $O(\log n)$ states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with $O(\log\log n)$ states. We give essentially matching lower bounds for the class of 1-aware protocols.