Monadic Presburger Predicates have Robust Population Protocols

TL;DR

This paper proves that all predicates in monadic Presburger arithmetic have robust population protocols and analyzes the state complexity cost of robustness, showing it can be double exponential.

Contribution

It extends robustness results to monadic Presburger predicates and evaluates the state complexity cost of robustness in population protocols.

Findings

All monadic Presburger predicates have robust protocols.

The cost of robustness can be at least double exponential in predicate size.

Lossin et al.'s protocols for threshold predicates are state-optimal.

Abstract

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.