The Black Ninjas and the Sniper: On Robustness of Population Protocols

TL;DR

This work investigates robustness of population protocols under adversarial silent crashes (snipers) and shows that while the classic threshold protocol for predicates like $x \ge t$ can be fragile, robust variants exist for certain predicates. The authors formalize a generalized protocol framework with snipers, define $i$-executions and fairness, and construct robust protocols for threshold and modulo predicates, while demonstrating that robustness is not preserved under standard Boolean combinations. They connect robustness to Presburger predicates and discuss open questions about universal robustness, offering a partial positive answer and a formalism that supports fault-tolerant distributed computation in population protocols. The results have implications for designing fault-tolerant distributed systems where agents can silently fail during interactions.

Abstract

Population protocols are a model of distributed computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs to decide some property of their initial configuration. We investigate the behaviour of population protocols under adversarial faults that cause agents to silently crash and no longer interact with other agents. As a starting point, we consider the property ``the number of agents exceeds a given threshold $t$'', represented by the predicate $x \geq t$, and show that the standard protocol for $x \geq t$ is very fragile: one single crash in a computation with $x:=2t-1$ agents can already cause the protocol to answer incorrectly that $x \geq t$ does not hold. However, a slightly less known protocol is robust: for any number $t' \geq t$ of agents, at least $t' - t+1$ crashes must occur for the protocol to answer that the property does not hold. We formally define robustness for arbitrary population protocols, and investigate the question whether every predicate computable by population protocols has a robust protocol. Angluin et al. proved in 2007 that population protocols decide exactly the Presburger predicates, which can be represented as Boolean combinations of threshold predicates of the form $\sum_{i=1}^n a_i \cdot x_i \geq t$ for $a_1,...,a_n, t \in \mathbb{Z}$ and modulo prdicates of the form $\sum_{i=1}^n a_i \cdot x_i \bmod m \geq t $ for $a_1, \ldots, a_n, m, t \in \mathbb{N}$. We design robust protocols for all threshold and modulo predicates. We also show that, unfortunately, the techniques in the literature that construct a protocol for a Boolean combination of predicates given protocols for the conjuncts do not preserve robustness. So the question remains open.

Figures (1)

• Figure 1: Petri nets for the first (left) and second (right) protocols with attack threshold 4. Accepting states are shaded in blue, rejecting states in red. The number of pebbles (left) or the level (right) of a state are written next to its corresponding place in boldface. The tokens in the places show configurations reachable from the initial configuration with $5$ ninjas. The accumulation transitions are colored in blue.

Theorems & Definitions (1)

• Example 1