Lower Bounds on the State Complexity of Population Protocols

TL;DR

This paper establishes the first non-trivial lower bounds on the state complexity of population protocols, showing that certain predicates require at least logarithmic or non-elementary states, thus advancing understanding of their computational limits.

Contribution

It provides the first non-trivial lower bounds on the state complexity of population protocols for specific predicates, complementing previous upper bounds.

Findings

Lower bound of (\u00f7 \u00f8) for leaderless protocols

Lower bound of inverse non-elementary function for protocols with leaders

Advances understanding of the minimal states needed for population protocol computation

Abstract

Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin \textit{et al.} has shown that the counting predicates $x \geq \eta$ have state complexity $\mathcal{O}(\log \eta)$ for leaderless protocols and $\mathcal{O}(\log \log \eta)$ for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of $x \geq \eta$ is $\Omega(\log\log \eta)$ for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.