A Faster P Solution for the Byzantine Agreement Problem

  • Michael J. Dinneen
  • Yun-Bum Kim
  • Radu Nicolescu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6501)


We propose an improved generic version of P modules, an extensible framework for recursive composition of P systems. We further provide a revised P solution for the Byzantine agreement problem, based on Exponential Information Gathering (EIG) trees, for N processes connected in a complete graph. Each process is modelled by the combination of N + 1 modules: one “main” module, plus one “firewall” communication module for each process (including one for itself). The EIG tree evaluation functionality is localized into a “main” single cell P module. The messaging functionality is localized into a three cells communication P module. This revised P solution improves overall running time from 9L + 6 to 6L + 1, where L is the number of messaging rounds. Most of the running time, 5L steps, is spent on the communication overhead. We briefly discuss if single cells can solve the Byzantine agreement without support and protection from additional communication cells; we conjecture that this is not possible, within the currently accepted definitions.


P systems P modules Byzantine agreement Distributed algorithms Modular design 


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  1. 1.
    Abd-El-Malek, M., Ganger, G.R., Goodson, G.R., Reiter, M.K., Wylie, J.J.: Fault-scalable Byzantine fault-tolerant services. In: Herbert, A., Birman, K.P. (eds.) SOSP, pp. 59–74. ACM, New York (2005)Google Scholar
  2. 2.
    Ben-Or, M., Hassidim, A.: Fast quantum Byzantine agreement. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 481–485. ACM, New York (2005)Google Scholar
  3. 3.
    Cachin, C., Kursawe, K., Shoup, V.: Random oracles in constantinople: Practical asynchronous Byzantine agreement using cryptography. J. Cryptology 18(3), 219–246 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castro, M., Liskov, B.: Practical Byzantine fault tolerance and proactive recovery. ACM Trans. Comput. Syst. 20(4), 398–461 (2002)CrossRefGoogle Scholar
  5. 5.
    Ciobanu, G.: Distributed algorithms over communicating membrane systems. Biosystems 70(2), 123–133 (2003)CrossRefGoogle Scholar
  6. 6.
    Ciobanu, G., Desai, R., Kumar, A.: Membrane systems and distributed computing. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) WMC 2002. LNCS, vol. 2597, pp. 187–202. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Dinneen, M.J., Kim, Y.-B., Nicolescu, R.: A faster P solution for the Byzantine agreement problem. Report CDMTCS-388, Centre for Discrete Mathematics and Theoretical Computer Science, The University of Auckland, Auckland (July 2010)Google Scholar
  8. 8.
    Dinneen, M.J., Kim, Y.-B., Nicolescu, R.: P systems and the Byzantine agreement. The Journal of Logic and Algebraic Programming (in Press, 2010) Corrected Proof)Google Scholar
  9. 9.
    Froehlich, F.E., Kent, A.: Encyclopedia of Telecommunications, vol. 15. CRC Press, Boca Raton (1997)Google Scholar
  10. 10.
    Lamport, L., Shostak, R.E., Pease, M.C.: The Byzantine generals problem. ACM Trans. Program. Lang. Syst. 4(3), 382–401 (1982)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann Publishers Inc., San Francisco (1996)zbMATHGoogle Scholar
  12. 12.
    Martin, J.-P., Alvisi, L.: Fast Byzantine consensus. IEEE Trans. Dependable Sec. Comput. 3(3), 202–215 (2006)CrossRefGoogle Scholar
  13. 13.
    Nicolescu, R., Dinneen, M.J., Kim, Y.-B.: Towards structured modelling with hyperdag P systems. International Journal of Computers, Communications and Control 2, 209–222 (2010)Google Scholar
  14. 14.
    Păun, G.: Membrane Computing: An Introduction. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Păun, G.: Introduction to membrane computing. In: Ciobanu, G., Pérez-Jiménez, M.J., Păun, G. (eds.) Applications of Membrane Computing. Natural Computing Series, pp. 1–42. Springer, Heidelberg (2006)Google Scholar
  16. 16.
    Păun, G., Pérez-Jiménez, M.J.: Solving problems in a distributed way in membrane computing: dP systems. International Journal of Computers, Communications and Control 5(2), 238–252 (2010)CrossRefGoogle Scholar
  17. 17.
    Pease, M.C., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Romero-Campero, F.J., Twycross, J., Cámara, M., Bennett, M., Gheorghe, M., Krasnogor, N.: Modular assembly of cell systems biology models using P systems. Int. J. Found. Comput. Sci. 20(3), 427–442 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Serbanuta, T., Stefanescu, G., Rosu, G.: Defining and executing P systems with structured data in K. In: Corne, D.W., Frisco, P., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2008. LNCS, vol. 5391, pp. 374–393. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michael J. Dinneen
    • 1
  • Yun-Bum Kim
    • 1
  • Radu Nicolescu
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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