Skip to main content
Springer Nature Link
Log in

Zero-Information Protocols and Unambiguity in Arthur–Merlin Communication

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur–Merlin (\({{\textsf {AM}}}\)) communication protocols. Our starting point is to show that—in contrast to plain randomized communication complexity—every boolean function admits an \({{\textsf {AM}}}\) communication protocol where on each yes-input, the distribution of Merlin’s proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur’s randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward \({{\textsf {AM}}}\). Zero-information protocols (\({{\textsf {ZAM}}}\)): Our basic \({{\textsf {ZAM}}}\) protocol uses exponential communication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes \({{\textsf {NL}}}\) and \({\oplus }{{{\textsf {L}}}}\) have polynomial-communication \({{\textsf {ZAM}}}\) protocols. We also prove that \({{\textsf {ZAM}}}\) complexity is lower bounded by conondeterministic communication complexity. Unambiguous protocols (\({{\textsf {UAM}}}\)): Our most technically substantial result is a \(\Omega (n)\) lower bound on the \({{\textsf {UAM}}}\) complexity of the \({{\textsf {NP}}}\)-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the “zero-information barrier” described above. We also prove that in general, \({{\textsf {UAM}}}\) complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Ada, A., Chattopadhyay, A., Cook, S., Fontes, L., Koucký, M., Pitassi, T.: The hardness of being private. ACM Trans. Comput. Theory (2014). doi:10.1145/2567671

    MathSciNet  MATH  Google Scholar 

  2. Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC\(^{\text{0 }}\). SIAM J. Comput. 36(4), 845–888 (2006). doi:10.1137/S0097539705446950

    Article  MathSciNet  MATH  Google Scholar 

  3. Aaronson, S., Wigderson, A.: Algebrization: a new barrier in complexity theory. ACM Trans. Comput. Theory (2009). doi:10.1145/1490270.1490272

    MATH  Google Scholar 

  4. Braverman, M., Ellen, F., Oshman, R., Pitassi, T., Vaikuntanathan, V.: A tight bound for set disjointness in the message-passing model. In: Proceedings of the 54th Symposium on Foundations of Computer Science (FOCS), pp. 668–677. IEEE (2013). doi:10.1109/FOCS.2013.77

  5. Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pp. 337–347. IEEE (1986). doi:10.1109/SFCS.1986.15

  6. Böhler, E., Glaßer, C., Meister, D.: Error-bounded probabilistic computations between MA and AM. J. Comput. Syst. Sci. 72(6), 1043–1076 (2006). doi:10.1016/j.jcss.2006.05.001

    Article  MathSciNet  MATH  Google Scholar 

  7. Braverman, M., Garg, A., Pankratov, D., Weinstein, O.: From information to exact communication. In: Proceedings of the 45th Symposium on Theory of Computing (STOC), pp. 151–160. ACM (2013). doi:10.1145/2488608.2488628

  8. Babai, L., Moran, S.: Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci. 36(2), 254–276 (1988). doi:10.1016/0022-0000(88)90028-1

    Article  MathSciNet  MATH  Google Scholar 

  9. Braverman, M., Moitra, A.: An information complexity approach to extended formulations. In: Proceedings of the 45th Symposium on Theory of Computing (STOC), pp. 161–170. ACM (2013). doi:10.1145/2488608.2488629

  10. Beigel, R., Reingold, N., Spielman, D.: PP is closed under intersection. J. Comput. Syst. Sci. 50(2), 191–202 (1995). doi:10.1006/jcss.1995.1017

    Article  MathSciNet  MATH  Google Scholar 

  11. Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004). doi:10.1016/j.jcss.2003.11.006

    Article  MathSciNet  MATH  Google Scholar 

  12. Chakrabarti, A., Cormode, G., Goyal, N., Thaler, J.: Annotations for sparse data streams. In: Proceedings of the 25th Symposium on Discrete Algorithms (SODA), pp. 687–706. ACM-SIAM (2014). doi:10.1137/1.9781611973402.52

  13. Chakrabarti, A., Cormode, G., McGregor, A., Thaler, J., Venkatasubramanian, S.: Verifiable stream computation and Arthur–Merlin communication. In: Proceedings of the 30th Computational Complexity Conference (CCC), pp. 217–243. Schloss Dagstuhl (2015). doi:10.4230/LIPIcs.CCC.2015.217

  14. Chakrabarti, A., Cormode, G., McGregor, A., Thaler, J.: Annotations in data streams. ACM Trans. Algorithms 11(1), 7 (2014). doi:10.1145/2636924

    Article  MathSciNet  MATH  Google Scholar 

  15. Chakrabarti, A., Khot, S., Sun, X.: Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In: Proceedings of the 18th Conference on Computational Complexity (CCC), pp. 107–117. IEEE (2003). doi:10.1109/CCC.2003.1214414

  16. Chakrabarti, A., Shi, Y., Wirth, A., Yao, A.: Informational complexity and the direct sum problem for simultaneous message complexity. In: Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS), pp. 270–278. IEEE (2001). doi:10.1109/SFCS.2001.959901

  17. Damm, C.: Problems complete for \(\oplus \)L. Inf. Process. Lett. 36(5), 247–250 (1990). doi:10.1016/0020-0190(90)90150-V

    Article  MathSciNet  MATH  Google Scholar 

  18. Dasgupta, A., Kumar, R., Sivakumar, D.: Sparse and lopsided set disjointness via information theory. In: Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pp. 517–528. Springer (2012). doi:10.1007/978-3-642-32512-0_44

  19. Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation. In: Proceedings of the 26th Symposium on Theory of Computing (STOC), pp. 554–563. ACM (1994). doi:10.1145/195058.195408

  20. Göös, M., Lovett, S., Meka, R.: Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. In: Proceedings of the 47th Symposium on Theory of Computing (STOC), pp. 257–266. ACM (2015). doi:10.1145/2746539.2746596

  21. Gur, T., Raz, R.: Arthur-Merlin streaming complexity. Inf. Comput. 243, 145–165 (2015). doi:10.1016/j.ic.2014.12.011

    Article  MathSciNet  MATH  Google Scholar 

  22. Gur, T., Rothblum, R.: Non-interactive proofs of proximity. In: Proceedings of the 6th Innovations in Theoretical Computer Science Conference (ITCS), pp. 133–142. ACM (2015). doi:10.1145/2688073.2688079

  23. Gronemeier, A.: Asymptotically optimal lower bounds on the NIH-multi-party information complexity of the AND-function and disjointness. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 505–516. Schloss Dagstuhl (2009). doi:10.4230/LIPIcs.STACS.2009.1846

  24. Gavinsky, D., Sherstov, A.: A separation of NP and coNP in multiparty communication complexity. Theory Comput. 6(1), 227–245 (2010). doi:10.4086/toc.2010.v006a010

    Article  MathSciNet  MATH  Google Scholar 

  25. Grolmusz, V., Tardos, G.: A note on non-deterministic communication complexity with few witnesses. Theory Comput. Syst. 36(4), 387–391 (2003). doi:10.1007/s00224-003-1158-7

    Article  MathSciNet  MATH  Google Scholar 

  26. Gál, A., Wigderson, A.: Boolean complexity classes vs. their arithmetic analogs. Random Struct. Algorithms 9(1–2), 99–111 (1996). doi:10.1002/(SICI)1098-2418(199608/09)9:1/2<99::AID-RSA7>3.0.CO;2-6

  27. Göös, M., Watson, T.: Communication complexity of set-disjointness for all probabilities. In: Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM), pp. 721–736. Schloss Dagstuhl (2014). doi:10.4230/LIPIcs.APPROX-RANDOM.2014.721

  28. Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Proceedings of the 29th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 244–256. Springer (2002). doi:10.1007/3-540-45465-9_22

  29. Impagliazzo, R., Kabanets, V., Kolokolova, A.: An axiomatic approach to algebrization. In: Proceedings of the 41st Symposium on Theory of Computing (STOC), pp. 695–704. ACM (2009). doi:10.1145/1536414.1536509

  30. Impagliazzo, R., Williams, R.: Communication complexity with synchronized clocks. In: Proceedings of the 25th Conference on Computational Complexity (CCC), pp. 259–269. IEEE (2010). doi:10.1109/CCC.2010.32

  31. Jayram, T.S.: Hellinger strikes back: a note on the multi-party information complexity of AND. In: Proceedings of the 13th International Workshop on Randomization and Computation (RANDOM), pp. 562–573. Springer (2009). doi:10.1007/978-3-642-03685-9_42

  32. Jain, R., Klauck, H.: The partition bound for classical communication complexity and query complexity. In: Proceedings of the 25th Conference on Computational Complexity (CCC), pp. 247–258. IEEE (2010). doi:10.1109/CCC.2010.31

  33. Johnson, N., Kemp, A., Kotz, S.: Univariate Discrete Distributions, 3rd edn. Wiley, New York (2005)

    Book  MATH  Google Scholar 

  34. Jayram, T.S., Kumar, R., Sivakumar, D.: Two applications of information complexity. In: Proceedings of the 35th Symposium on Theory of Computing (STOC), pp. 673–682. ACM (2003). doi:10.1145/780542.780640

  35. Jukna, S.: On graph complexity. Comb. Probab. Comput. 15(6), 855–876 (2006). doi:10.1017/S0963548306007620

    Article  MathSciNet  MATH  Google Scholar 

  36. Jukna, S.: Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, Berlin (2012)

  37. Klauck, H.: Rectangle size bounds and threshold covers in communication complexity. In: Proceedings of the 18th Conference on Computational Complexity (CCC), pp. 118–134. IEEE (2003). doi:10.1109/CCC.2003.1214415

  38. Klauck, H.: Lower bounds for quantum communication complexity. SIAM J. Comput. 37(1), 20–46 (2007). doi:10.1137/S0097539702405620

    Article  MathSciNet  MATH  Google Scholar 

  39. Klauck, H.: A strong direct product theorem for disjointness. In: Proceedings of the 42nd Symposium on Theory of Computing (STOC), pp. 77–86. ACM (2010). doi:10.1145/1806689.1806702

  40. Klauck, H.: On Arthur Merlin games in communication complexity. In: Proceedings of the 26th Conference on Computational Complexity (CCC), pp. 189–199. IEEE (2011). doi:10.1109/CCC.2011.33

  41. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  42. Karchmer, M., Newman, I., Saks, M., Wigderson, A.: Non-deterministic communication complexity with few witnesses. J. Comput. Syst. Sci. 49(2), 247–257 (1994). doi:10.1016/S0022-0000(05)80049-2

    Article  MathSciNet  MATH  Google Scholar 

  43. Klauck, H., Prakash, V.: Streaming computations with a loquacious prover. In: Proceedings of the 4th Innovations in Theoretical Computer Science Conference (ITCS), pp. 305–320. ACM (2013). doi:10.1145/2422436.2422471

  44. Klauck, H., Podder, S.: Two results about quantum messages. In: Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 445–456. Springer (2014). doi:10.1007/978-3-662-44465-8_38

  45. Klauck, H., Prakash, V.: An improved interactive streaming algorithm for the distinct elements problem. In: Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), pp. 919–930. Springer (2014). doi:10.1007/978-3-662-43948-7_76

  46. Kushilevitz, E.: Privacy and communication complexity. SIAM J. Discrete Math. 5(2), 273–284 (1992). doi:10.1137/0405021

    Article  MathSciNet  MATH  Google Scholar 

  47. Lin, J.: Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 37(1), 145–151 (1991). doi:10.1109/18.61115

    Article  MathSciNet  MATH  Google Scholar 

  48. Lokam, S.: Spectral methods for matrix rigidity with applications to size-depth trade-offs and communication complexity. J. Comput. Syst. Sci. 63(3), 449–473 (2001). doi:10.1006/jcss.2001.1786

    Article  MathSciNet  MATH  Google Scholar 

  49. Lokam, S.: Complexity lower bounds using linear algebra. Found. Trends Theor. Comput. Sci. 4(1–2), 1–155 (2009). doi:10.1561/0400000011

    MathSciNet  MATH  Google Scholar 

  50. Linial, N., Shraibman, A.: Learning complexity versus communication complexity. Comb. Probab. Comput. 18(1–2), 227–245 (2009). doi:10.1017/S0963548308009656

    Article  MathSciNet  MATH  Google Scholar 

  51. Pudlák, P., Rödl, V., Savický, P.: Graph complexity. Acta Inform. 25(5), 515–535 (1988). doi:10.1007/BF00279952

    MathSciNet  MATH  Google Scholar 

  52. Papakonstantinou, P., Scheder, D., Song, H.: Overlays and limited memory communication. In: Proceedings of the 29th Conference on Computational Complexity (CCC), pp. 298–308. IEEE (2014). doi:10.1109/CCC.2014.37

  53. Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM J. Comput. 29(4), 1118–1131 (2000). doi:10.1137/S0097539798339041

    Article  MathSciNet  MATH  Google Scholar 

  54. Razborov, A.: On rigid matrices. Technical report, Steklov Mathematical Institute (1989) (In Russian)

  55. Razborov, A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992). doi:10.1016/0304-3975(92)90260-M

    Article  MathSciNet  MATH  Google Scholar 

  56. Raz, R., Shpilka, A.: On the power of quantum proofs. In: Proceedings of the 19th Conference on Computational Complexity (CCC), pp. 260–274. IEEE (2004). doi:10.1109/CCC.2004.1313849

  57. Santha, M.: Relativized Arthur-Merlin versus Merlin-Arthur games. Inf. Comput. 80(1), 44–49 (1989). doi:10.1016/0890-5401(89)90022-9

    Article  MathSciNet  MATH  Google Scholar 

  58. Schöning, U.: Probabilistic complexity classes and lowness. J. Comput. Syst. Sci. 39(1), 84–100 (1989). doi:10.1016/0022-0000(89)90020-2

    Article  MathSciNet  MATH  Google Scholar 

  59. Valiant, L.: Graph-theoretic arguments in low-level complexity. In: Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 162–176. Springer (1977). doi:10.1007/3-540-08353-7_135

  60. Valiant, L.: Completeness classes in algebra. In: Proceedings of the 11th Symposium on Theory of Computing (STOC), pp. 249–261. ACM (1979). doi:10.1145/800135.804419

  61. Wunderlich, H.: A note on a problem in communication complexity. Technical report, arXiv (2012). arXiv:1205.0903

  62. Wunderlich, H.: On a theorem of Razborov. Comput. Complex. 21(3), 431–477 (2012). doi:10.1007/s00037-011-0021-5

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank Petteri Kaski, Venkatesh Medabalimi, and Robert Robere for discussions and anonymous reviewers for comments. Supported by funding from NSERC.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Toronto, Canada

    Mika Göös, Toniann Pitassi & Thomas Watson

Authors
  1. Mika Göös
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Toniann Pitassi
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Thomas Watson
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Thomas Watson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Göös, M., Pitassi, T. & Watson, T. Zero-Information Protocols and Unambiguity in Arthur–Merlin Communication. Algorithmica 76, 684–719 (2016). https://doi.org/10.1007/s00453-015-0104-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-015-0104-9

Keywords