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Synchronous Gathering Without Multiplicity Detection: A Certified Algorithm

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Stabilization, Safety, and Security of Distributed Systems (SSS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10083))

Abstract

In mobile robotic swarms, the gathering problem consists in coordinating all the robots so that in finite time they occupy the same location, not known beforehand. Multiplicity detection refers to the ability to detect that more than one robot can occupy a given position. When the robotic swarm operates synchronously, a well-known result by Cohen and Peleg permits to achieve gathering, provided robots are capable of multiplicity detection.

We present a new algorithm for synchronous gathering, that does not assume that robots are capable of multiplicity detection, nor make any other extra assumption. Unlike previous approaches, our proof correctness is certified in the model where the protocol is defined, using the Coq proof assistant.

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Notes

  1. 1.

    The 2016 SIROCCO Prize for Innovation in Distributed Computing was awarded to Masafumi Yamashita for this line of work.

  2. 2.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR.Impossibility.html.

  3. 3.

    http://coq.inria.fr.

  4. 4.

    http://coq.inria.fr.

  5. 5.

    Available at http://pactole.lri.fr.

  6. 6.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Robots.html#Robots.

  7. 7.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Configurations.html#Configuration.

  8. 8.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Configurations.html#Spectrum.

  9. 9.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.FlexibleFormalism.html.

  10. 10.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.CommonFormalism.html#Sig.robogram.

  11. 11.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.FlexDefinitions.html#FlexGatheringDefs.gathered_at.

  12. 12.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.FlexDefinitions.html#FlexGatheringDefs.FullSolGathering.

  13. 13.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.ffgatherR2_pgm.

  14. 14.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.R2geometry.html#barycenter.

  15. 15.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.round_simplify.

  16. 16.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.max_dist_spect_ex.

  17. 17.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.round_last_step.

  18. 18.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.round_lt_config.

  19. 19.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.gathered_at_OK.

  20. 20.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.not_barycenter_moves.

  21. 21.

    http://pactole.lri.fr/pub/cffg2d/html/Pactole.Gathering.InR2.FSyncFlexNoMultAlgorithm.html#GatheringinR2.FSGathering_in_R2.

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Acknowledgements

The authors are grateful to the reviewers who provided constructive comments and helped to improve the presentation of this work.

Author information

Authors and Affiliations

  1. Collège de France, 75006, Paris, France

    Lionel Rieg

  2. École Nat. Sup. d’Informatique Pour l’Industrie et l’Entreprise (ENSIIE), 91025, Évry, France

    Amélie Delga & Xavier Urbain

  3. LRI, CNRS UMR 8623, Université Paris-Sud, Université Paris-Saclay, Orsay, France

    Thibaut Balabonski & Xavier Urbain

  4. UPMC Sorbonne Universités, LIP6-CNRS 7606, Paris, France

    Amélie Delga & Sébastien Tixeuil

  5. Institut Universitaire de France, Paris, France

    Sébastien Tixeuil

Authors
  1. Thibaut Balabonski
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  2. Amélie Delga
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  3. Lionel Rieg
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  4. Sébastien Tixeuil
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  5. Xavier Urbain
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Corresponding author

Correspondence to Xavier Urbain .

Editor information

Editors and Affiliations

  1. McMaster University, Hamilton, Ontario, Canada

    Borzoo Bonakdarpour

  2. LIP6, INRIA, UPMC Sorbonne Universities, Paris, France

    Franck Petit

A Axioms of the Formalisation

A Axioms of the Formalisation

In the main file , the last command:

shows all the axioms upon which the proof of correctness of our algorithm for gathering in \(\mathbb {R} ^2\) relies, in total 31 axioms. Here, we break them down. They can be classified in three categories:

  • The first category is the axiomatisation of reals numbers from the Coq standard library. It is by far the biggest number of axioms, and they are not listed here.

  • The second category is the description of the problem.

    figure q

    As one can see, it simply means that our proof is valid for any number  of robots greater than or equal to 2. Notice that with one robot or less, the problem is not interesting (trivially solved).

  • The third category contains usual geometric properties that are not part of our library: firstly some properties about barycenters that we think could be provable from its axiomatisation but are currently left as axioms, that the barycenter is unique and the result of the function computing the barycenter is indeed a barycenter:

    figure r

    Finally that similarities can be expressed with an orthogonal matrix, a zoom factor and a translation. The orthogonal matrix and the scaling factor are combined into two column vectors and .

    figure s

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Balabonski, T., Delga, A., Rieg, L., Tixeuil, S., Urbain, X. (2016). Synchronous Gathering Without Multiplicity Detection: A Certified Algorithm. In: Bonakdarpour, B., Petit, F. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2016. Lecture Notes in Computer Science(), vol 10083. Springer, Cham. https://doi.org/10.1007/978-3-319-49259-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-49259-9_2

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  • Online ISBN: 978-3-319-49259-9

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