Sound 3-Query PCPPs Are Long

  • Eli Ben-Sasson
  • Prahladh Harsha
  • Oded Lachish
  • Arie Matsliah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3-query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the proof-length and show that any verifier obtaining “best possible” soundness must query an exponentially long proof.


Query Complexity Good Soundness Checkable Proof Oracle Access Unique Game 
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. [ALM+98]
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal ACM 45(3), 501–555 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. [AS98]
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal ACM 45(1), 70–122 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. [BCH+96]
    Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M.A., Sudan, M.: Linearity testing in characteristic two. IEEE Transactions on Information Theory 42(6), 1781–1795 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. [BGH+06]
    Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. SICOMP 36(4), 889–974 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. [BHLM07]
    Ben-Sasson, E., Harsha, P., Lachish, O., Matsliah, A.: Sound 3-query PCPPs are long. Technical Report TR07-127, ECCC (2007)Google Scholar
  6. [BS05]
    Ben-Sasson, E., Sudan, M.: Simple PCPs with poly-log rate and query complexity. In: Proc. 37th ACM STOC, Baltimore, Maryland, pp. 266–275.Google Scholar
  7. [Din07]
    Dinur, I.: The PCP theorem by gap amplification. Journal ACM 54(3), 12 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. [DR06]
    Dinur, I., Reingold, O.: Assignment testers: Towards a combinatorial proof of the PCP Theorem. SICOMP 36, 975–1024 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. [EKR04]
    Ergün, F., Kumar, R., Rubinfeld, R.: Fast approximate probabilistically checkable proofs. Information and Computation 189(2), 135–159 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. [FF05]
    Fischer, E., Fortnow, L.: Tolerant versus intolerant testing for boolean properties. In: Proc. 20th IEEE CCC, San Jose, California, pp. 135–140.Google Scholar
  11. [Fis01]
    Fischer, E.: The art of uninformed decisions: A primer to property testing. Bulletin of the European Association for Theoretical Computer Science 75, 97–126 (2001); The Computational Complexity ColumnMathSciNetMATHGoogle Scholar
  12. [GGR98]
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal ACM 45(4), 653–750 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. [GR05]
    Guruswami, V., Rudra, A.: Tolerant locally testable codes. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 306–317. Springer, Heidelberg (2005)Google Scholar
  14. [LR99]
    Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal ACM 46(6), 787–832 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. [PRR06]
    Parnas, M., Ron, D., Rubinfeld, R.: Tolerant property testing and distance approximation. Journal of Computer and System Sciences 72(6), 1012–1042 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. [Sze99]
    Szegedy, M.: Many-valued logics and holographic proofs. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 676–686. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. [Tre05]
    Trevisan, L.: Approximation algorithms for unique games. In: Proc. 46th IEEE FOCS, Pittsburgh, Pennsylvania, pp. 197–205Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Eli Ben-Sasson
    • 1
  • Prahladh Harsha
    • 2
  • Oded Lachish
    • 3
  • Arie Matsliah
    • 1
  1. 1.Computer Science DepartmentTechnion, Israel Institute of TechnologyHaifaIsrael
  2. 2.Toyota Technological InstituteChicagoUSA
  3. 3.Centre for Discrete Mathematics and its Applications (DIMAP)University of WarwickCoventryUnited Kingdom

Personalised recommendations