Monadic Presburger Predicates have Robust Population Protocols

2026-04-30

AI Summary

This work investigates the design of robust population protocols resilient to adversarial crash failures for univariate Presburger predicates in a model of anonymous finite-state agents interacting via random pairwise encounters. By integrating population protocol theory, Presburger arithmetic representations, and state complexity analysis, the study establishes—for the first time—that every univariate Presburger predicate admits a robust protocol. It further demonstrates that robustness incurs a double-exponential lower bound on state complexity relative to the predicate’s size. Additionally, the paper proves that the threshold-predicate protocol proposed by Lossin et al. achieves optimal state complexity, matching this lower bound.

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.

Problem

• population protocols
• robustness
• Presburger arithmetic
• crash failures
• monadic predicates

Innovation

• robust population protocols
• monadic Presburger arithmetic
• state complexity
• crash failures
• distributed consensus