The Expressive Power of Uniform Population Protocols with Logarithmic Space

TL;DR

This work resolves the expressiveness gap for population protocols with $Θ(f(n))$ states, showing that, for reasonable $f(n)$ with $f(n) ∈ Ω(\log n) ∩ O(n^{1-ε})$, uniform and weakly uniform protocols can decide exactly those predicates whose unary encoding lies in $NSPACE( f(n) \log n )$, i.e. $\mathsf{UPP}(f(n))=\mathsf{WUPP}(f(n))=\mathsf{UNSPACE}(f(n)\log n)$. The core technique is a lower-bound construction that simulates a $O(f(n)\log n)$-space nondeterministic TM using only $O(f(n))$ states, achieved via a multi-phase protocol that distributes the population size, encodes counters with digits, implements zero-checks, and orchestrates a counter machine simulation. An accompanying upper bound follows from standard reductions and Immerman–Szelepcsényi-style arguments, yielding a tight characterization and a corollary for $f(n)=\log n$ giving $\mathsf{UNSPACE}(\log^2 n)$. These results complete the time-space tradeoff landscape for polylogarithmic-state population protocols and have implications for how such protocols map to space-bounded computation, with potential future work on time-restricted models and decidability boundaries.

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 $Ω(n)$ states. This leaves a significant gap, particularly concerning protocols with $Θ(\log n)$ or $Θ(\mathsf{polylog}~ n)$ states, which are the most common constructions in the literature. In this paper we close the gap and prove that for any $ε > 0$ and $f {\in}Ω(\log n) {\cap}O(n^{1-ε})$, both uniform and non-uniform population protocols with $Θ(f(n))$ states can decide exactly those predicates, whose unary encoding lies in $\mathsf{NSPACE}(f(n) \log n)$.

Theorems & Definitions (12)

• Definition 1
• Example 2
• Example 3
• Definition 4
• Example 5
• Definition 6
• Theorem 7
• Proposition 7
• Theorem 8
• Lemma 8