Distributed Computing

, Volume 20, Issue 4, pp 279–304 | Cite as

The computational power of population protocols

  • Dana Angluin
  • James Aspnes
  • David Eisenstat
  • Eric Ruppert
Article

Abstract

We consider the model of population protocols introduced by Angluin et al. (Computation in networks of passively mobile finite-state sensors, pp. 290–299. ACM, New York, 2004), in which anonymous finite-state agents stably compute a predicate of the multiset of their inputs via two-way interactions in the family of all-pairs communication networks. We prove that all predicates stably computable in this model (and certain generalizations of it) are semilinear, answering a central open question about the power of the model. Removing the assumption of two-way interaction, we also consider several variants of the model in which agents communicate by anonymous message-passing where the recipient of each message is chosen by an adversary and the sender is not identified to the recipient. These one-way models are distinguished by whether messages are delivered immediately or after a delay, whether a sender can record that it has sent a message, and whether a recipient can queue incoming messages, refusing to accept new messages until it has had a chance to send out messages of its own. We characterize the classes of predicates stably computable in each of these one-way models using natural subclasses of the semilinear predicates.

Keywords

Input Symbol Fairness Condition Population Protocol Presburger Arithmetic Fair Execution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Angluin, D., Aspnes, J., Chan, M., Fischer, M.J., Jiang, H., Peralta, R.: Stably computable properties of network graphs. In: Prasanna, V.K., Iyengar, S., Spirakis, P., Welsh, M. (eds) Distributed Computing in Sensor Systems: First IEEE International Conference, DCOSS 2005, Marina del Rey, CA, USA, June/July, 2005, Proceedings, volume 3560 of Lecture Notes in Computer Science, pp. 63–74. Springer, Berlin (2005)Google Scholar
  2. 2.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Urn automata. Technical Report YALEU/DCS/TR-1280, Yale University Department of Computer Science (2003)Google Scholar
  3. 3.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. In: PODC ’04: Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, pp. 290–299. ACM, New York (2004)Google Scholar
  4. 4.
    Angluin D., Aspnes J., Diamadi Z., Fischer M.J. and Peralta R. (2006). Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18(4): 235–253 CrossRefGoogle Scholar
  5. 5.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. In: Distributed Computing: 20th International Symposium, DISC 2006: Stockholm, Sweden, September 2006: Proceedings, pp. 61–75 (2006)Google Scholar
  6. 6.
    Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: Proceedings of the 25th ACM Symposium on Principles of Distributed Computing, pp. 292–299 (2006)Google Scholar
  7. 7.
    Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: On the power of anonymous one-way communication. In: Principles of Distributed Systems; 9th International Conference, OPODIS 2005; Pisa, Italy; December 2005; Revised Selected Papers, volume 3974 of Lecture Notes in Computer Science, pp. 396–411 (2005)Google Scholar
  8. 8.
    Angluin, D., Aspnes, J., Fischer, M.J., Jiang, H.: Self-stabilizing population protocols. In: Principles of Distributed Systems; 9th International Conference, OPODIS 2005; Pisa, Italy; December 2005; Revised Selected Papers, volume 3974 of Lecture Notes in Computer Science. pp. 103–117 (2005)Google Scholar
  9. 9.
    Angluin, D., Fischer, M.J., Jiang, H.: Stabilizing consensus in mobile networks. In: Proceedings of Distributed Computing in Sensor Systems: Second IEEE International Conference, volume 4026 of Lecture Notes in Computer Science, pp. 37–50 (2006)Google Scholar
  10. 10.
    Attiya H., Gorbach A. and Moran S. (2002). Computing in totally anonymous asynchronous shared memory systems. Inform. Comput. 173(2): 162–183 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 82–93 (1980)Google Scholar
  12. 12.
    Aspnes, J., Shah, G., Shah, J.: Wait-free consensus with infinite arrivals. In: Proceedings of the 34th ACM Symposium on Theory of Computing, pp. 524–533 (2002)Google Scholar
  13. 13.
    Bailey N.T.J. (1975). The Mathematical Theory of Infectious Diseases, Second Edition. Charles Griffin & Co., London Google Scholar
  14. 14.
    Buhrman H. (2006). Alessandro Panconesi, Riccardo Silvestri and Paul Vitanyi. On the importance of having an identity or, is consensus really universal?. Distrib. Comput. 18(3): 167–176 CrossRefGoogle Scholar
  15. 15.
    Boldi, P., Vigna, S.: Computing anonymously with arbitrary knowledge. In: Proceedings of the 18th ACM Symposium on Principles of Distributed Computing, pp. 173–179 (1999)Google Scholar
  16. 16.
    Boldi, P., Vigna, S.: An effective characterization of computability in anonymous networks. In: Distributed Computing, 15th International Conference, pp. 33–47 (2001)Google Scholar
  17. 17.
    Diamadi, Z., Fischer, M.J.: A simple game for the study of trust in distributed systems. Wuhan Univ. J. Natural Sci. 6(1–2), 72–82 (2001). Also appears as Yale Technical Report TR–1207, January 2001Google Scholar
  18. 18.
    Delporte-Gallet, C., Fauconnier, H., Guerraoui, R., Ruppert, E.: When birds die: Making population protocols fault-tolerant. In: Proceedings of the Second IEEE International Conference on Distributed Computing in Sensor Systems (DCOSS ’06), pp. 51–66 (2006)Google Scholar
  19. 19.
    Dickson L.E. (1913). Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Am. J. Math. 35(4): 413–422 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Daley D.J. and Kendall D.G. (1965). Stochastic rumours. J. Inst. Math. Appl. 1: 42–55 CrossRefMathSciNetGoogle Scholar
  21. 21.
    Eğecioğlu Ö. and Singh A.K. (1994). Naming symmetric processes using shared variables. Distrib. Comput. 8(1): 19–38 CrossRefGoogle Scholar
  22. 22.
    Fischer, M.J., Jiang, H.: Self-stabilizing leader election in networks of finite-state anonymous agents. In: Tenth International Conference on Principles of Distributed Systems, volume 4305 of Lecture Notes in Computer Science, pp. 395–409 (2006)Google Scholar
  23. 23.
    Fich F. and Ruppert E. (2003). Hundreds of impossibility results for distributed computing. Distrib. Comput. 16(2–3): 121–163 CrossRefGoogle Scholar
  24. 24.
    Gibson M.A. and Bruck J. (2000). Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104: 1876–1880 CrossRefGoogle Scholar
  25. 25.
    Gillespie D.T. (1992). A rigorous derivation of the chemical master equation. Physica A 188: 404–425 CrossRefGoogle Scholar
  26. 26.
    Guerraoui, R., Ruppert, E.: Anonymous and fault-tolerant shared-memory computing. Distrib. Comput. (2007, in press)Google Scholar
  27. 27.
    Higman G. (1952). Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 3(2): 326–336 CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hopcroft J. and Pansiot J. (1978). On the reachability problem for 5-dimensional vector addition systems. Theoret. Comput. Sci. 8(2): 135–159 CrossRefMathSciNetGoogle Scholar
  29. 29.
    Ibarra O.H., Dang Z. and Egecioglu O. (2004). Catalytic P systems, semilinear sets, and vector addition systems. Theor. Comput. Sci. 312(2–3): 379–399 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Jiang, H.: Distributed Systems of Simple Interacting Agents. PhD thesis, Yale University (2007)Google Scholar
  31. 31.
    Jayanti, P., Toueg, S.: Wakeup under read/write atomicity. In: Distributed Algorithms, 4th International Workshop, volume 486 of LNCS, pp. 277–288 (1990)Google Scholar
  32. 32.
    Kutten S., Ostrovsky R. and Patt-Shamir B. (2000). The Las-Vegas processor identity problem (How and when to be unique). J. Algorithms 37(2): 468–494 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Lang S. (2002). Algebra (revised third edition). Springer, Berlin Google Scholar
  34. 34.
    Lay, S.R.: Convex Sets and their Applications. Krieger Publishing Company, (1992)Google Scholar
  35. 35.
    Lipton R.J. and Park A. (1990). The processor identity problem. Inform. Process. Lett. 36(2): 91–94 MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Parikh R.J. (1966). On context-free languages. J. ACM 13(4): 570–581 MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Panconesi A., Papatriantafilou M., Tsigas P. and Vitányi P. (1998). Randomized naming using wait-free shared variables. Distrib. Comput. 11(3): 113–124 CrossRefGoogle Scholar
  38. 38.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes-Rendus du I Congrès de Mathématiciens des Pays Slaves, pp. 92–101, Warszawa (1929)Google Scholar
  39. 39.
    Sakamoto, N.: Comparison of initial conditions for distributed algorithms on anonymous networks. In: Proc. 18th ACM Symposium on Principles of Distributed Computing, pp. 173–179 (1999)Google Scholar
  40. 40.
    Teng S.-H. (1990). Space efficient processor identity protocol. Inform. Process. Lett. 34(3): 147–154 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Dana Angluin
    • 1
  • James Aspnes
    • 1
  • David Eisenstat
    • 2
  • Eric Ruppert
    • 3
  1. 1.Yale UniversityNew HavenUSA
  2. 2.Princeton UniversityPrincetonUSA
  3. 3.York UniversityTorontoCanada

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