Languages with Bounded Multiparty Communication Complexity

  • Arkadev Chattopadhyay
  • Andreas Krebs
  • Michal Koucký
  • Mario Szegedy
  • Pascal Tesson
  • Denis Thérien
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


We study languages with bounded communication complexity in the multiparty “input on the forehead model” with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0.

In contrast, we use coding techniques to show that there are languages of arbitrarily large circuit complexity which can be recognized in constant communication by k players for k ≥ 3. However, we show that if a language has a neutral letter and bounded communication complexity in the k-party game for some fixed k then the language is in fact regular. We give an algebraic characterization of regular languages with this property. We also prove that a symmetric language has bounded k-party complexity for some fixed k iff it has bounded two party complexity.


Word Problem Communication Complexity Regular Language Commutative Monoids Symmetric Boolean Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambainis, A.: Upper bounds on multiparty communication complexity of shifts. In: Proc. 13th STACS, pp. 631–642 (1996)Google Scholar
  2. 2.
    Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. JCSS 45(2), 204–232 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barrington, D.A.M., et al.: First order expressibility of languages with neutral letters or: The Crane Beach conjecture. JCSS 70(2), 101–127 (2005)zbMATHGoogle Scholar
  4. 4.
    Barrington, D.A.M., Straubing, H.: Superlinear lower bounds for bounded-width branching programs. JCSS 50(3), 374–381 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beame, P., Pitassi, T., Segerlind, N.: Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity. In: Caires, L., et al. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1176–1188. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Beame, P., Vee, E.: Time-space tradeoffs multiparty communication complexity and nearest neighbor problems. In: 34th STOC, pp. 688–697 (2002)Google Scholar
  7. 7.
    Chandra, A.K., Furst, M.L., Lipton, R.J.: Multi-party protocols. In: STOC’83, pp. 94–99 (1983)Google Scholar
  8. 8.
    Chattopadhyay, A., et al.: Languages with bounded multiparty communication complexity. In: ECCC, TR06-118 (2006)Google Scholar
  9. 9.
    Goldmann, M., Håstad, J.: Monotone circuits for connectivity have depth (log n)2 − o(1). SIAM J. Comput. 27(5), 1283–1294 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Graham, R.L., Rotschild, B.L., Spencer, J.H.: Ramsey Theorey. Series in Discrete Mathematics. Wiley Interscience, Chichester (1980)Google Scholar
  11. 11.
    Grolmusz, V.: Separating the communication complexities of MOD m and MOD p circuits. In: Proc. 33rd FOCS, pp. 278–287 (1992)Google Scholar
  12. 12.
    Grolmusz, V.: The BNS lower bound for multi-party protocols in nearly optimal. Information and Computation 112(1), 51–54 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  14. 14.
    Nisan, N.: The communication complexity of treshold gates. In: Combinatorics, Paul Erdös is Eighty, vol. 1, pp. 301–315 (1993)Google Scholar
  15. 15.
    Pin, J.-E.: Syntactic semigroups. In: Handbook of language theory, vol. 1, pp. 679–746. Springer, Heidelberg (1997)Google Scholar
  16. 16.
    Pudlák, P.: An application of Hindman’s theorem to a problem on communication complexity. Combinatorics, Probability and Computing 12(5–6), 661–670 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Raymond, J.-F., Tesson, P., Thérien, D.: An algebraic approach to communication complexity. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 29–40. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  18. 18.
    Raz, R.: The BNS-Chung criterion for multi-party communication complexity. Computational Complexity 9(2), 113–122 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Szegedy, M.: Functions with bounded symmetric communication complexity, programs over commutative monoids, and ACC. JCSS 47(3), 405–423 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tesson, P., Thérien, D.: Complete classifications for the communication complexity of regular languages. Theory of Computing Systems 38(2), 135–159 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Tesson, P., Thérien, D.: Restricted two-variable sentences, circuits and communication complexity. In: Caires, L., et al. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 526–538. Springer, Heidelberg (2005)Google Scholar
  22. 22.
    Thérien, D.: Subword counting and nilpotent groups. In: Combinatorics on Words: Progress and Perspectives, pp. 195–208. Academic Press, London (1983)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Arkadev Chattopadhyay
    • 1
  • Andreas Krebs
    • 2
  • Michal Koucký
    • 3
  • Mario Szegedy
    • 4
  • Pascal Tesson
    • 5
  • Denis Thérien
    • 1
  1. 1.School of Computer Science, McGill University, Montreal 
  2. 2.Universität TübingenGermany
  3. 3.Mathematical Institute, Academy of SciencesCzech Republic
  4. 4.Rutgers University, New Jersey 
  5. 5.Laval University, Québec 

Personalised recommendations