Population protocols are a model of distributed computation in which a collection of indistinguishable finite-state agents interact randomly in pairs to decide a predicate of their initial configuration. The agents decide by achieving a stable consensus on whether the predicate holds or not. It is known that population protocols can decide exactly the predicates expressible in Presburger arithmetic. Recently, Lossin et al. have introduced a notion of protocol robustness against adversarial crash failures. They show that all atomic Presburger predicates can be decided by robust protocols, and ask whether the same holds for every Presburger predicate. We make progress towards settling this question by proving that all predicates expressible in monadic Presburger arithmetic have robust protocols. In addition, we analyze the cost of robustness in terms of state complexity. We study the ratio between the number of states of the smallest robust protocol for a given predicate and the smallest protocol for it. We show that the cost of robustness is at least double exponential in the size of the predicate, and prove that the robust protocols by Lossin et al. for threshold predicates x ≥ k have optimal state complexity.

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