Brief Announcement: Population Protocols Decide Double-exponential Thresholds

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 φn is succinct if it uses O(|φn|) states, where φn 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(|φn|) states exist for any family of predicates φn. We answer this affirmatively, by constructing protocols with O(log|φn|) states for some family of threshold predicates φn(x) ⇔ x ≥ kn, with k1, k2, ... ∈ ℕ. (In other words, protocols with O(n) states that decide x ≥ k for a k ≥ 2n.) This matches a known lower bound. Moreover, our construction is the first that is not 1-aware, making it robust against some types of errors.