Synchronous Gathering without Multiplicity Detection: a Certified Algorithm

  • Thibaut Balabonski
  • Amélie Delga
  • Lionel Rieg
  • Sébastien Tixeuil
  • Xavier Urbain
Article
First Online:
  • 247 Downloads
Part of the following topical collections:
  1. Special Issue on Stabilization, Safety, and Security of Distributed Systems (SSS 2016)

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, the correctness of our proof is certified in the model where the protocol is defined, using the Coq proof assistant.

Keywords

Mobile robots Gathering Multiplicity detection Certification Proof assistant 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

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

References

  1. 1.
    Altisen, K., Corbineau, P., Devismes, S.: A framework for certified self-stabilization. In: Albert, E, Lanese, I (eds.) Formal Techniques for Distributed Objects, Components, and Systems - 36th IFIP WG 6.1 International Conference, FORTE 2016, Held as Part of the 11th International Federated Conference on Distributed Computing Techniques, DisCoTec 2016, Heraklion, Crete, Greece, June 6-9, 2016, Proceedings, volume 9688 of Lecture Notes in Computer Science, pp. 36–51. Springer (2016)Google Scholar
  2. 2.
    Auger, C., Bouzid, Z., Courtieu, P., Tixeuil, S., Urbain, X.: Certified impossibility results for byzantine-tolerant mobile robots. In: Higashino, T, Katayama, Y, Masuzawa, T, Potop-Butucaru, M, Yamashita, M (eds.) Stabilization, Safety, and Security of Distributed Systems - 15th International Symposium (SSS 2013), volume 8255 of Lecture Notes in Computer Science, pp. 178–186. Springer, Osaka (2013)Google Scholar
  3. 3.
    Balabonski, T., Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Certified gathering of oblivious mobile robots: Survey of recent results and open problems. In: Petrucci, L, Seceleanu, C, Cavalcanti, A (eds.) Critical Systems: Formal Methods and Automated Verification - Joint 22nd International Workshop on Formal Methods for Industrial Critical Systems - and - 17th International Workshop on Automated Verification of Critical Systems, (FMICS-AVoCS 2017), volume 10471 of Lecture Notes in Computer Science, pp. 165–181. Springer, Turin (2017)Google Scholar
  4. 4.
    Balabonski, T., Pelle, R., Rieg, L., Tixeuil, S.: A foundational framework for certified impossibility results with mobile robots on graphs. In: Proceedings of International Conference on Distributed Computing and Networking. Varanasi (2018)Google Scholar
  5. 5.
    Bérard, B., Lafourcade, P., Millet, L., Potop-Butucaru, M., Thierry-Mieg, Y., Tixeuil, S.: Formal verification of mobile robot protocols. Distrib. Comput. 29(6), 459–487 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bertot, Y., Castéran, P.: Interactive theorem proving and program development. Coq’Art: The calculus of inductive constructions. Texts in Theoretical Computer Science. Springer (2004)Google Scholar
  7. 7.
    Bonnet, F., Dėfago, X., Petit, F., Potop-Butucaru, M., Tixeuil, S.: Discovering and assessing fine-grained metrics in robot networks protocols. In: 33rd IEEE International Symposium on Reliable Distributed Systems Workshops, SRDS Workshops 2014, pp. 50–59. IEEE, Nara (2014)Google Scholar
  8. 8.
    Bérard, B., Courtieu, P., Millet, L., Potop-Butucaru, M., Rieg, L., Sznajder, N., Tixeuil, S., Urbain, X.: Formal methods for mobile robots: Current results and open problems. Int. J. Inf. Soc. 7(3), 101–114 (2015). Invited PaperGoogle Scholar
  9. 9.
    Castėran, P., Filou, V.: Tasks, types and tactics for local computation systems. Studia Informatica Universalis 9(1), 39–86 (2011)Google Scholar
  10. 10.
    Cohen, R., Peleg, D.: Robot convergence via center-of-gravity algorithms. In: Kralovic, R, Sýkora, O (eds.) Structural Information and Communication Complexity - 11th International Colloquium (SIROCCO 2004), volume 3104 of Lecture Notes in Computer Science, pp. 79–88. Springer, Smolenice Castle (2004)Google Scholar
  11. 11.
    Cohen, R., Peleg, D.: Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput. 34(6), 1516–1528 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Coquand, T., Paulin-Mohring, C.: Inductively defined types. In: Martin-Löf, P, Mints, G (eds.) International Conference on Computer Logic (Colog’88), volume 417 of Lecture Notes in Computer Science, pp. 50–66. Springer (1990)Google Scholar
  13. 13.
    Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Impossibility of gathering, a certification. Inf. Process. Lett. 115, 447–452 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Certified universal gathering algorithm in \(\mathbb {R}^{2}\) for oblivious mobile robots. In: Gavoille, C, Ilcinkas, D (eds.) Distributed Computing - 30th International Symposium, (DISC 2016), volume 9888 of Lecture Notes in Computer Science. Springer, Paris (2016)Google Scholar
  15. 15.
    Devismes, S., Lamani, A., Petit, F., Raymond, P., Tixeuil, S.: Optimal Grid Exploration by Asynchronous Oblivious Robots. In: Richa, A W, Scheideler, C (eds.) Stabilization, Safety, and Security of Distributed Systems - 14th International Symposium (SSS 2012), volume 7596 of Lecture Notes in Computer Science, pp. 64–76. Springer, Toronto (2012)Google Scholar
  16. 16.
    Doan, HTT., Bonnet, F., Ogata, K.: Model checking of robot gathering. In: Aspnes, J., Felber, P. (eds.) Principles of Distributed Systems - 21th International Conference (OPODIS 2017), Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Lisbon (2017)Google Scholar
  17. 17.
    Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers (2012)Google Scholar
  18. 18.
    Millet, L., Potop-Butucaru, M., Sznajder, N., Tixeuil, S.: On the synthesis of mobile robots algorithms: The case of ring gathering. In: Felber, P., Garg, V.K. (eds.) Stabilization, Safety, and Security of Distributed Systems - 16th International Symposium, (SSS 2014), volume 8756 of Lecture Notes in Computer Science, pp. 237–251. Springer, Paderborn (2014)Google Scholar
  19. 19.
    Prencipe, G.: Impossibility of gathering by a set of autonomous mobile robots. Theor. Comput. Sci. 384(2-3), 222–231 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rubin, S., Zuleger, F., Murano, A., Aminof, B.: Verification of asynchronous mobile-robots in partially-known environments. In: Chen, Q., Torroni, P., Villata, S., Hsu, J.Y.-j., Omicini, A. (eds.) PRIMA 2015: Principles and Practice of Multi-Agent Systems - 18th International Conference, Bertinoro, Italy, October 26-30, 2015, Proceedings, volume 9387 of Lecture Notes in Computer Science, pp. 185–200. Springer (2015)Google Scholar
  21. 21.
    Sangiorgi, D.: Introduction to Bisimulation and Coinduction. Cambridge University Press (2012)Google Scholar
  22. 22.
    Sangnier, A., Sznajder, N., Potop-Butucaru, M., Tixeuil, S.: Parameterized verification of algorithms for oblivious robots on a ring. In: Formal Methods in Computer Aided Design. Vienna (2017)Google Scholar
  23. 23.
    Suzuki, I., Yamashita, M.: Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.LRI, Université Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance
  2. 2.École Nationale Supérieure d’Informatique pour l’Industrie et l’Entreprise (ENSIIE)ÉvryFrance
  3. 3.UPMC Sorbonne UniversitésParisFrance
  4. 4.Collège de FranceParisFrance
  5. 5.Yale UniversityNew HavenUSA
  6. 6.Institut Universitaire de FranceParisFrance
  7. 7.Université Claude Bernard Lyon-1, LIRIS CNRS UMR 5205Université de LyonLyonFrance

Personalised recommendations