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Breaking Through the Ω(n)-Space Barrier: Population Protocols Decide Double-Exponential Thresholds

Author Philipp Czerner

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Author Details

Philipp Czerner
  • Department of Informatics, TU München, Germany

Funding

  • Czerner, Philipp: This work was supported by an ERC Advanced Grant (787367: PaVeS) and by the Research Training Network of the Deutsche Forschungsgemeinschaft (DFG) (378803395: ConVeY).

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Philipp Czerner. Breaking Through the Ω(n)-Space Barrier: Population Protocols Decide Double-Exponential Thresholds. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.DISC.2024.17

Abstract

Population protocols are a model of distributed computation in which finite-state agents interact randomly in pairs. A protocol decides for any initial configuration whether it satisfies a fixed property, specified as a predicate on the set of configurations. A family of protocols deciding predicates φ_n is succinct if it uses 𝒪(|φ_n|) states, where φ_n is encoded as quantifier-free Presburger formula with coefficients in binary. (All predicates decidable by population protocols can be encoded in this manner.) While it is known that succinct protocols exist for all predicates, it is open whether protocols with o(|φ_n|) states exist for any family of predicates φ_n. We answer this affirmatively, by constructing protocols with 𝒪(log|φ_n|) states for some family of threshold predicates φ_n(x) ⇔ x ≥ k_n, with k₁,k₂,... ∈ ℕ. (In other words, protocols with 𝒪(n) states that decide x ≥ k for a k ≥ 2^2ⁿ.) This matches a known lower bound. Moreover, our construction for threshold predicates is the first that is not 1-aware, and it is almost self-stabilising.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed computing models
Keywords
  • Distributed computing
  • population protocols
  • state complexity

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