skip to main content
10.1145/3583668.3594571acmconferencesArticle/Chapter ViewAbstractPublication PagespodcConference Proceedingsconference-collections
short-paper

Brief Announcement: Population Protocols Decide Double-exponential Thresholds

Published: 16 June 2023 Publication History

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 φn is succinct if it uses O(|φn|) states, where φn 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(|φn|) states exist for any family of predicates φn. We answer this affirmatively, by constructing protocols with O(log|φn|) states for some family of threshold predicates φn(x) ⇔ xkn, with k1, k2, ... ∈ ℕ. (In other words, protocols with O(n) states that decide xk for a k ≥ 2n.) This matches a known lower bound. Moreover, our construction is the first that is not 1-aware, making it robust against some types of errors.

References

[1]
Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. 2004. Computation in networks of passively mobile finite-state sensors. In PODC. ACM, 290--299.
[2]
Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. 2006. Computation in networks of passively mobile finite-state sensors. Distributed Comput. 18, 4 (2006), 235--253.
[3]
Dana Angluin, James Aspnes, David Eisenstat, and Eric Ruppert. 2007. The computational power of population protocols. Distributed Comput. 20, 4 (2007), 279--304.
[4]
Dana Angluin, James Aspnes, Michael J. Fischer, and Hong Jiang. 2005. Self-stabilizing Population Protocols. In Principles of Distributed Systems, 9th International Conference, OPODIS 2005, Pisa, Italy, December 12--14, 2005, Revised Selected Papers (Lecture Notes in Computer Science, Vol. 3974), James H. Anderson, Giuseppe Prencipe, and Roger Wattenhofer (Eds.). Springer, 103--117.
[5]
Michael Blondin, Javier Esparza, Blaise Genest, Martin Helfrich, and Stefan Jaax. 2020. Succinct Population Protocols for Presburger Arithmetic. In STACS (LIPIcs, Vol. 154). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 40:1--40:15.
[6]
Michael Blondin, Javier Esparza, and Stefan Jaax. 2018. Large Flocks of Small Birds: On the Minimal Size of Population Protocols. In STACS (LIPIcs, Vol. 96). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 16:1--16:14.
[7]
Janna Burman, Ho-Lin Chen, Hsueh-Ping Chen, David Doty, Thomas Nowak, Eric E. Severson, and Chuan Xu. 2021. Time-Optimal Self-Stabilizing Leader Election in Population Protocols. In PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26--30, 2021, Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen (Eds.). ACM, 33--44.
[8]
Shukai Cai, Taisuke Izumi, and Koichi Wada. 2012. How to Prove Impossibility Under Global Fairness: On Space Complexity of Self-Stabilizing Leader Election on a Population Protocol Model. Theory Comput. Syst. 50, 3 (2012), 433--445.
[9]
Ho-Lin Chen, Rachel Cummings, David Doty, and David Soloveichik. 2017. Speed faults in computation by chemical reaction networks. Distributed Comput. 30, 5 (2017), 373--390.
[10]
Philipp Czerner. 2022. Leaderless Population Protocols Decide Double-exponential Thresholds. CoRR abs/2204.02115 (2022). arXiv:2204.02115
[11]
Philipp Czerner and Javier Esparza. 2021. Lower Bounds on the State Complexity of Population Protocols. In PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26--30, 2021, Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen (Eds.). ACM, 45--54.
[12]
Philipp Czerner, Javier Esparza, and Jérôme Leroux. 2021. Lower Bounds on the State Complexity of Population Protocols. CoRR (2021). arXiv:2102.11619v3
[13]
Moez Draief and Milan Vojnovic. 2012. Convergence Speed of Binary Interval Consensus. SIAM J. Control. Optim. 50, 3 (2012), 1087--1109.
[14]
Jérôme Leroux. 2022. State Complexity of Protocols with Leaders. In PODC '22: ACM Symposium on Principles of Distributed Computing, Salerno, Italy, July 25 -- 29, 2022, Alessia Milani and Philipp Woelfel (Eds.). ACM, 257--264.
[15]
Richard J. Lipton. 1976. The Reachability Problem Requires Exponential Space. Technical Report. Yale University, Dept. of CS. http://www.cs.yale.edu/publications/techreports/tr63.pdf
[16]
Etienne Perron, Dinkar Vasudevan, and Milan Vojnovic. 2009. Using Three States for Binary Consensus on Complete Graphs. In INFOCOM 2009. 28th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 19--25 April 2009, Rio de Janeiro, Brazil. IEEE, 2527--2535.

Cited By

View all
  • (2024)Selective Population ProtocolsStabilization, Safety, and Security of Distributed Systems10.1007/978-3-031-74498-3_16(225-239)Online publication date: 20-Oct-2024
  • (2022)State Complexity of Protocols with LeadersProceedings of the 2022 ACM Symposium on Principles of Distributed Computing10.1145/3519270.3538421(257-264)Online publication date: 21-Jul-2022

Index Terms

  1. Brief Announcement: Population Protocols Decide Double-exponential Thresholds

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Conferences
    PODC '23: Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing
    June 2023
    392 pages
    ISBN:9798400701214
    DOI:10.1145/3583668
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 16 June 2023

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. distributed computing
    2. population protocols
    3. space complexity

    Qualifiers

    • Short-paper

    Funding Sources

    Conference

    PODC '23
    Sponsor:

    Acceptance Rates

    PODC '23 Paper Acceptance Rate 29 of 110 submissions, 26%;
    Overall Acceptance Rate 740 of 2,477 submissions, 30%

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)14
    • Downloads (Last 6 weeks)0
    Reflects downloads up to 20 Feb 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2024)Selective Population ProtocolsStabilization, Safety, and Security of Distributed Systems10.1007/978-3-031-74498-3_16(225-239)Online publication date: 20-Oct-2024
    • (2022)State Complexity of Protocols with LeadersProceedings of the 2022 ACM Symposium on Principles of Distributed Computing10.1145/3519270.3538421(257-264)Online publication date: 21-Jul-2022

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media