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

16 December 2024
Benno Lossin
P. Czerner
Javier Esparza
Roland Guttenberg
Tobias Prehn
ArXiv (abs)PDFHTML
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 ttt'', represented by the predicate x≥tx \geq tx≥t, and show that the standard protocol for x≥tx \geq tx≥t is very fragile: one single crash in a computation with x:=2t−1x:=2t-1x:=2t−1 agents can already cause the protocol to answer incorrectly that x≥tx \geq tx≥t does not hold. However, a slightly less known protocol is robust: for any number t′≥tt' \geq tt′≥t of agents, at least t′−t+1t' - t+1t′−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 ∑i=1nai⋅xi≥t\sum_{i=1}^n a_i \cdot x_i \geq t∑i=1n​ai​⋅xi​≥t for a1,...,an,t∈Za_1,...,a_n, t \in \mathbb{Z}a1​,...,an​,t∈Z and modulo prdicates of the form ∑i=1nai⋅xi mod m≥t\sum_{i=1}^n a_i \cdot x_i \bmod m \geq t ∑i=1n​ai​⋅xi​modm≥t for a1,…,an,m,t∈Na_1, \ldots, a_n, m, t \in \mathbb{N}a1​,…,an​,m,t∈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.

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