On Distributed Merlin-Arthur Decision Protocols

  • Pierre Fraigniaud
  • Pedro MontealegreEmail author
  • Rotem Oshman
  • Ivan Rapaport
  • Ioan Todinca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)


In a distributed locally-checkable proof, we are interested in checking the legality of a given network configuration with respect to some Boolean predicate. To do so, the network enlists the help of a prover—a computationally-unbounded oracle that aims at convincing the network that its state is legal, by providing the nodes with certificates that form a distributed proof of legality. The nodes then verify the proof by examining their certificate, their local neighborhood and the certificates of their neighbors.

In this paper we examine the power of a randomized form of locally-checkable proof, called distributed Merlin-Arthur protocols, or \({\textsf {dMA}}\) for short. In a \({\textsf {dMA}}\) protocol, the prover assigns each node a short certificate, and the nodes then exchange random messages with their neighbors. We show that while there exist problems for which \({\textsf {dMA}}\) protocols are more efficient than protocols that do not use randomness, for several natural problems, including Leader Election, Diameter, Symmetry, and Counting Distinct Elements, \({\textsf {dMA}}\) protocols are no more efficient than standard nondeterministic protocols. This is in contrast with Arthur-Merlin (\({\textsf {dAM}}\)) protocols and Randomized Proof Labeling Schemes (RPLS), which are known to provide improvements in certificate size, at least for some of the aforementioned properties.


Distributed verification Nondeterminism Interactive computation Interactive proof systems 



Partially supported by CONICYT PIA/Apoyo a Centros Científicos y Tecnológicos de Excelencia AFB 170001 (P.M. and I.R.), Fondecyt 1170021 (I.R.) and CONICYT via PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.). Rotem Oshman is supported by ISF i-core Center for Excellence, No. 4/11.


  1. 1.
    Agarwal, P.K., Cormode, G., Huang, Z., Phillips, J.M., Wei, Z., Yi, K.: Mergeable summaries. ACM Trans. Database Syst. 38(4), 26:1–26:28 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58(1), 137–147 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baruch, M., Fraigniaud, P., Patt-Shamir, B.: Randomized proof-labeling schemes. In: 34th ACM Symposium on Principles of Distributed Computing (PODC), pp. 315–324 (2015)Google Scholar
  4. 4.
    Censor-Hillel, K., Khoury, S., Paz, A.: Quadratic and near-quadratic lower bounds for the CONGEST model (2017). arXiv preprint arXiv:1705.05646
  5. 5.
    Censor-Hillel, K., Paz, A., Perry, M.: Approximate proof-labeling schemes. In: Das, S., Tixeuil, S. (eds.) SIROCCO 2017. LNCS, vol. 10641, pp. 71–89. Springer, Cham (2017). Scholar
  6. 6.
    Cook, S.A.: The Complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing (STOC 1971), New York, NY, USA, pp. 151–158. ACM (1971)Google Scholar
  7. 7.
    Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Hungar. 14(3–4), 295–315 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fraigniaud, P., Korman, A., Peleg, D.: Towards a complexity theory for local distributed computing. J. ACM 60(5), 35:1–35:26 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Flajolet, P., Martin, G.N.: Probabilistic counting algorithms for data base applications. J. Comput. Syst. Sci. 31(2), 182–209 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. J. ACM (JACM) 38(3), 690–728 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Göös, M., Suomela, J.: Locally checkable proofs in distributed computing. Theory Comput. 12(1), 1–33 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kushilevitz, E., Nisan, N.: Communication Complexity, pp. 1–189. Cambridge University Press, New York (1997). ISBN 978-0-521-56067-2zbMATHGoogle Scholar
  14. 14.
    Kol, G., Oshman, R., Saxena, R.R.: Interactive distributed proofs. In: 37th ACM Symposium on Principles of Distributed Computing (PODC), pp. 255–264 (2018)Google Scholar
  15. 15.
    Kol, G., Oshman, R., Saxena, R.R.: AM Lower Bound for Symmetry. Private communication (2019)Google Scholar
  16. 16.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22(4), 215–233 (2010). Scholar
  17. 17.
    Kushilevitz, E., Nissan, N.: Communication Complexity. Cambridge University Press, Cambridge (2006)Google Scholar
  18. 18.
    Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. J. ACM 39(4), 859–868 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Naor, M., Parter, M., Yogev, E.: The power of distributed verifiers in interactive proofs (2018). CoRR abs/1812.10917Google Scholar
  20. 20.
    Naor, M., Stockmeyer, L.J.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Patt-Shamir, B.: A note on efficient aggregate queries in sensor networks. Theor. Comput. Sci. 370(1–3), 254–264 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Shamir, A.: IP = PSPACE. J. ACM 39(4), 869–877 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Pierre Fraigniaud
    • 1
  • Pedro Montealegre
    • 2
    Email author
  • Rotem Oshman
    • 3
  • Ivan Rapaport
    • 4
  • Ioan Todinca
    • 5
  1. 1.CNRS and Université de ParisParisFrance
  2. 2.Universidad Adolfo IbáñezSantiagoChile
  3. 3.Tel-Aviv UniversityTel Aviv-YafoIsrael
  4. 4.DIM-CMM (UMI 2807 CNRS)Universidad de ChileSantiagoChile
  5. 5.Université d’OrléansOrléansFrance

Personalised recommendations