The Expressive Power of Uniform Population Protocols with Logarithmic Space

2024-08-19

arXiv.org

AI Summary

This paper resolves a long-standing expressiveness gap in population protocols: while it was known that protocols with $o(log n)$ states decide exactly the semilinear predicates and those with $Omega(n)$ states characterize PSPACE, the computational power of protocols with $Theta(log n)$ to $mathrm{polylog}(n)$ states—the most practically relevant regime—remained uncharacterized. The authors provide the first exact characterization for uniform and non-uniform protocols with $Theta(f(n))$ states, where $f in Omega(log n) cap O(n^{1-varepsilon})$: such protocols decide precisely those predicates whose unary encodings correspond to languages in $mathbf{NSPACE}(f(n)log n)$. Technically, the result combines protocol normalization, Turing machine simulation, combinatorial constructions, and fine-grained space complexity analysis. This work unifies and substantially advances prior fragmented understanding, establishing an intrinsic equivalence between logarithmic-state population protocols and nondeterministic logarithmic-space computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $mathsf{NSPACE}(f(n) log n)$.

Problem

• Characterizing expressive power of logarithmic space population protocols
• Determining computational limits of uniform protocols with Θ(f(n)) states
• Closing the gap between o(log n) and Ω(n) state protocols

Innovation

• Uniform population protocols with logarithmic space states
• Protocols decide predicates in NSPACE with f(n) log n
• Closes gap between o(log n) and Ω(n) state protocols