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

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.