Breaking through the $Ω(n)$-space barrier: Population Protocols Decide Double-exponential Thresholds

TL;DR

This work addresses the Ω($n$)-space barrier for leaderless population protocols deciding threshold predicates by introducing population programs that enable Lipton-style double-exponential counting with error-detection. It proves the existence of leaderless protocols using $\mathcal{O}(n)$ states to decide thresholds $k\ge 2^{2^n}$, and shows these protocols are almost self-stabilising, providing robustness to arbitrary noisy initial configurations. The key methodology combines a structured programming model (population programs) with a two-stage transformation into population machines and then into population protocols, keeping state complexity near-optimal and enabling practical feasibility for chemical-like implementations. Overall, the paper closes the last major gap in threshold predicate space for leaderless population protocols and lays groundwork for further exploration of robust, succinct population computations.

Abstract

Population protocols are a model of distributed computation in which finite-state agents interact randomly in pairs. A protocol decides for any initial configuration whether it satisfies a fixed property, specified as a predicate on the set of configurations. A family of protocols deciding predicates $\varphi_n$ is succinct if it uses $\mathcal{O}(|\varphi_n|)$ states, where $\varphi_n$ is encoded as quantifier-free Presburger formula with coefficients in binary. (All predicates decidable by population protocols can be encoded in this manner.) While it is known that succinct protocols exist for all predicates, it is open whether protocols with $o(|\varphi_n|)$ states exist for \emph{any} family of predicates $\varphi_n$. We answer this affirmatively, by constructing protocols with $\mathcal{O}(\log|\varphi_n|)$ states for some family of threshold predicates $\varphi_n(x)\Leftrightarrow x\ge k_n$, with $k_1,k_2,...\in\mathbb{N}$. (In other words, protocols with $\mathcal{O}(n)$ states that decide $x\ge k$ for a $k\ge 2^{2^n}$.) This matches a known lower bound. Moreover, our construction for threshold predicates is the first that is not $1$-aware, and it is almost self-stabilising.

Figures (8)

• Figure 1: A population program for $\varphi(x)\Leftrightarrow 4\le x<7$ using registers $x,y,z$. Main is run initially and decides the predicate, $\Call{Test}{i}$ tries to move $i$ units from $x$ to $y$ and reports whether it succeeded, and Clean checks whether $z$ is empty and moves some number of units from $y$ to $x$. If Clean detects an agent in $z$, it restarts the computation. As every run calls Clean infinitely often, this serves to reject initial configurations where $z$ is nonzero; eventually the protocol will be restarted with $z=0$. This is an illustrative example and some simplifications are possible. E.g. the instruction $(\textbf{swap}\ x,y)$ in Clean is superfluous; additionally, instead of checking $z>0$ one could omit that register entirely.
• Figure 2: Example configurations exhibiting the different types.
• Figure 3: Conversion to a population machine.
• Figure 4: Converting instructions into transitions.
• Figure 5: Implementation of a while-loop.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 3
• Theorem 4
• Definition 5
• Definition 6
• Theorem 6
• Lemma 6
• Lemma 6