Decidability Classes for Mobile Agents Computing

  • Pierre Fraigniaud
  • Andrzej Pelc
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

We establish a classification of decision problems that are to be solved by mobile agents operating in unlabeled graphs, using a deterministic protocol. The classification is with respect to the ability of a team of agents to solve the problem, possibly with the aid of additional information. In particular, our focus is on studying differences between the decidability of a decision problem by agents and its verifiability when a certificate for a positive answer is provided to the agents. Our main result shows that there exists a natural complete problem for mobile agent verification. We also show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.

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References

  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM Journal on Computing 29, 1164–1188 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Angluin, D.: Local and global properties in networks of processors. In: STOC 1980, pp. 82–93 (1980)Google Scholar
  3. 3.
    Alpern, S.: Rendezvous search on labelled networks. Naval Research Logistics 49, 256–274 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alpern, S., Gal, S.: The theory of search games and rendezvous. Int. Series in Operations research and Management Science. Kluwer Academic Publisher (2002)Google Scholar
  5. 5.
    Alpern, J., Baston, V., Essegaier, S.: Rendezvous search on a graph. Journal of Applied Probability 36, 223–231 (1999)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Anderson, E., Fekete, S.: Asymmetric rendezvous on the plane. In: SoCG 1998, pp. 365–373 (1998)Google Scholar
  7. 7.
    Bender, M.A., Slonim, D.K.: The power of team exploration: Two robots can learn unlabeled directed graphs. In: FOCS 1994, pp. 75–85 (1994)Google Scholar
  8. 8.
    Boldi, P., Vigna, S.: Computing Anonymously with Arbitrary Knowledge. In: PODC 1999, pp. 181–188 (1999)Google Scholar
  9. 9.
    Boldi, P., Vigna, S.: An Effective Characterization of Computability in Anonymous Networks. In: Welch, J.L. (ed.) DISC 2001. LNCS, vol. 2180, pp. 33–47. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Boldi, P., Vigna, S.: Fibrations of graphs. Disc. Maths 243(1-3), 21–66 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chalopin, J., Das, S., Kosowski, A.: Constructing a Map of an Anonymous Graph: Applications of Universal Sequences. In: Lu, C., Masuzawa, T., Mosbah, M. (eds.) OPODIS 2010. LNCS, vol. 6490, pp. 119–134. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Chalopin, J., Godard, E., Métivier, Y.: Local Terminations and Distributed Computability in Anonymous Networks. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 47–62. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Solving the Robots Gathering Problem. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1181–1196. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Collins, A., Czyzowicz, J., Gąsieniec, L., Labourel, A.: Tell Me Where I Am So I Can Meet You Sooner. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 502–514. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: SODA 2010, pp. 22–30 (2010)Google Scholar
  16. 16.
    Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: Log-space rendezvous in arbitrary graphs. In: PODC 2010, pp. 450–459 (2010)Google Scholar
  17. 17.
    Das, S., Flocchini, P., Santoro, N., Yamashita, M.: On the computational power of oblivious robots: forming a series of geometric patterns. In: PODC 2010, pp. 267–276 (2010)Google Scholar
  18. 18.
    Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed Verification and Hardness of Distributed Approximation. In: STOC 2011, pp. 363–372 (2011)Google Scholar
  19. 19.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. Journal of Graph Theory 32, 265–297 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of Asynchronous Oblivious Robots with Limited Visibility. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 247–258. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Fraigniaud, P., Ilcinkas, D.: Digraphs Exploration with Little Memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Fraigniaud, P., Korman, A., Peleg, D.: Local Distributed Decision. In: FOCS 2011, pp. 708–717 (2011)Google Scholar
  24. 24.
    Fraigniaud, P., Gasieniec, L., Kowalski, D., Pelc, A.: Collective tree exploration. Networks 48, 166–177 (2006)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Fraigniaud, P., Pelc, A.: Deterministic Rendezvous in Trees with Little Memory. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 242–256. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  26. 26.
    Fraigniaud, P., Rajsbaum, S., Travers, C.: Locality and Checkability in Wait-free Computing. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 333–347. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  27. 27.
    Gasieniec, L., Pelc, A., Radzik, T., Zhang, X.: Tree exploration with logarithmic memory. In: SODA 2007, pp. 585–594 (2007)Google Scholar
  28. 28.
    Goos, M., Suomela, J.: Locally checkable proofs. In: PODC 2011, pp. 159–168 (2011)Google Scholar
  29. 29.
    Korman, A., Kutten, S., Peleg, D.: Proof Labeling Schemes. Distributed Computing 22, 215–233 (2010)CrossRefGoogle Scholar
  30. 30.
    Kowalski, D., Malinowski, A.: How to Meet in Anonymous Network. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 44–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Kranakis, E., Krizanc, D., Morin, P.: Randomized Rendez-Vous with Limited Memory. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 605–616. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  32. 32.
    Norris, N.: Universal covers of graphs: Isomorphism to depth n − 1 implies isomorphism to all depths. Discrete Applied Mathematics 56, 61–74 (1995)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Reingold, O.: Undirected connectivity in log-space. JACM 55, 1–24 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: SODA 2007, pp. 599–608 (2007)Google Scholar
  36. 36.
    Yamashita, M., Kameda, T.: Computing on Anonymous Networks: Part I-Characterizing the Solvable Cases. IEEE Trans. Par. Distrib. Syst. 7, 69–89 (1996)CrossRefGoogle Scholar
  37. 37.
    Yamashita, M., Kameda, T.: Computing Functions on Asynchronous Anonymous Networks. Mathematical Systems Theory 29(4), 331–356 (1996)MathSciNetMATHGoogle Scholar
  38. 38.
    Yamashita, M., Kameda, T.: Leader Election Problem on Networks in which Processor Identity Numbers Are Not Distinct. IEEE Trans. Parallel Distrib. Syst. 10(9), 878–887 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pierre Fraigniaud
    • 1
  • Andrzej Pelc
    • 2
  1. 1.CNRS and University Paris DiderotFrance
  2. 2.Université du Québec en OutaouaisCanada

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