State Complexity of Protocols With Leaders

TL;DR

This paper investigates the state complexity of population protocols with leaders, establishing a near-matching lower bound for the number of states needed to stably compute counting predicates, thus advancing understanding of their computational limits.

Contribution

It provides a new lower bound on the state complexity of counting predicates in protocols with leaders, refining previous bounds and contributing to the theoretical understanding of population protocol capabilities.

Findings

Established a lower bound matching the upper bound up to a square root.

Improved the inverse-Ackermannian lower bound from prior work.

Demonstrated the computational limits of protocols with leaders for counting predicates.

Abstract

Population protocols are a model of computation in which an arbitrary number of anonymous finite-memory agents are interacting in order to decide by stable consensus a predicate. In this paper, we focus on the counting predicates that asks, given an initial configuration, whether the number of agents in some initial state $i$ is at least $n$. In 2018, Blondin, Esparza, and Jaax shown that with a fix number of leaders and interaction-width, there exists infinitely many $n$ for which the counting predicate is stably computable by a protocol with at most $O(\log\log(n))$ states. We provide in this paper a matching lower-bound (up to a square root) that improves the inverse-Ackermannian lower-bound presented at PODC in 2021.