The Expressive Power of Uniform Population Protocols with Logarithmic Space

TL;DR

This paper characterizes the computational power of population protocols with logarithmic or polylogarithmic space, showing they can decide exactly the predicates in certain space-bounded classes, thus filling a key gap in the understanding of their expressive power.

Contribution

It proves that population protocols with Θ(f(n)) states can decide predicates in NSPACE(f(n) log n), for functions f between logarithmic and polynomial bounds, closing previous gaps in the field.

Findings

Protocols with Θ(f(n)) states decide predicates in NSPACE(f(n) log n).

The results apply to both uniform and non-uniform protocols.

This work extends the known expressive power of population protocols to logarithmic space bounds.

Abstract

Population protocols are a model of computation in which indistinguishable mobile agents interact in pairs to decide a property of their initial configuration. Originally introduced by Angluin et. al. in 2004 with a constant number of states, research nowadays focuses on protocols where the space usage depends on the number of agents. The expressive power of population protocols has so far however only been determined for protocols using $o(\log n)$ states, which compute only semilinear predicates, and for ${\Omega}(n)$ states. This leaves a significant gap, particularly concerning protocols with ${\Theta}(\log n)$ or ${\Theta}(\mathsf{polylog}~ n)$ states, which are the most common constructions in the literature. In this paper we close the gap and prove that for any ${\epsilon} > 0$ and $f {\in}{\Omega}(\log n) {\cap}O(n^{1-{\epsilon}})$, both uniform and non-uniform population protocols with ${\Theta}(f(n))$ states can decide exactly those predicates, whose unary encoding lies in $\mathsf{NSPACE}(f(n) \log n)$.