Population Protocols over Ordered Agents

2026-05-10

AI Summary

This work addresses the limitation of traditional anonymous population protocols in leveraging the ordering of agent identities by introducing the model PP[<] equipped with totally ordered identifiers and its immediate observation variant IO-PP[<], which restricts interactions to agents whose identifiers satisfy a given predicate, thereby enhancing expressive power. The paper establishes, for the first time, an exact correspondence between IO-PP[<] and unambiguous star-free languages, demonstrates that augmenting the model with a successor predicate elevates its computational power to NSPACE(n), and proves that consensus stability in PP[<] is undecidable. These results are achieved through a rigorous analysis grounded in formal language theory, finite automata, two-variable first-order logic, and space complexity, precisely characterizing the language classes recognizable, computational boundaries, and decidability thresholds of stability within these models.

Abstract

Population protocols are a distributed computation model in which a collection of anonymous, finite-state agents interact in randomly chosen pairs and update their states according to a fixed transition function. The computation is defined by the eventual stabilization of the population to a consensus that represents the output. In practice, it is natural to allow each agent to carry a unique identifier and compare it with that of another agent before interacting. We model this extension by having agents be totally ordered and interactions between two agents to be fireable only if their pair of identifiers falls in some condition set. For instance, $\mathsf{PP}[<]$ allows for two agents to interact only if the first one appears before the second one. We study population protocols over ordered agents $\mathsf{PP}[N]$ where $N$ is a set of predicates available to restrict transition firing. We also study $\textsf{IO-PP}[N]$, the immediate observation fragment of $\mathsf{PP}[N]$ where only one agent changes state per interaction. Our main result is that $\textsf{IO-PP}[<]$ recognizes exactly the unambiguous star-free languages, which admits many other characterizations, such as two-variable first-order logic or two-way deterministic partially-ordered automata. We also provide a logic and an automaton model that fits in $\mathsf{PP}[<]$. We further show that if the successor predicate appears in a set $N$ of $\mathsf{NSPACE}(n)$-computable predicates, then $\textsf{IO-PP}[N]=\mathsf{PP}[N]=\mathsf{NSPACE}(n)$. Finally, we investigate the problem of deciding whether a given population protocol always stabilizes to a consensus. While this problem is decidable for unordered population protocols, we show that this is undecidable already for $\mathsf{PP}[<]$ and $\textsf{IO-PP}[+1]$, but conditionally decidable for $\textsf{IO-PP}[<]$.

Problem

• Population Protocols
• Ordered Agents
• Consensus Stabilization
• Unambiguous Star-free Languages
• Decidability

Innovation

• Population Protocols
• Ordered Agents
• Immediate Observation
• Star-Free Languages
• Consensus Stabilization