Small PCPs with Low Query Complexity

  • Prahladh Harsha
  • Madhu Sudan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2010)


Most known constructions of probabilistically checkable proofs (PCPs) either blow up the proof size by a large polynomial, or have a high (though constant) query complexity. In this paper we give a transformation with slightly- super-cubic blowup in proof size, with a low query complexity. Specifically, the verifier probes the proof in 16 bits and rejects every proof of a false assertion with probability arbitrarily close to 1/2, while accepting corrects proofs of theo- rems with probability one. The proof is obtained by revisiting known construc- tions and improving numerous components therein. In the process we abstract a number of new modules that may be of use in other PCP constructions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., AND Szegedy, M. Proof verification and the hardness of approximation problems. Journal of the ACM 45, 3 (May 1998), 501–555.Google Scholar
  2. 2.
    Arora, S., AND Safra, S. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45, 1 (Jan. 1998), 70–122.Google Scholar
  3. 3.
    Arora, S., AND Sudan, M. Improved low degree testing and its applications. In Proc. 29th ACM Symp. on Theory of Computing (El Paso, Texas, 4–6 May 1997), pp. 485–495.Google Scholar
  4. 4.
    Bellare, M., Goldreich, O., AND Sudan, M. Free bits, PCPs, and nonapproximability—towards tight results. SIAM Journal of Computing 27, 3 (June 1998), 804–915.Google Scholar
  5. 5.
    Bellare, M., Goldwasser, S., Lund, C., AND Russell, A. Efficient probabilistically checkable proofs and applications to approximation. In Proc. 25th ACM Symp. on Theory of Computing (San Diego, California, 16–18 May 1993), pp. 294–304.Google Scholar
  6. 6.
    Cook, S. A. Short propositional formulas represent nondeterministic computations. Information Processing Letters 26, 5(11 Jan. 1988), 269–270.Google Scholar
  7. 7.
    Dinur, I., Fischer, E., Kindler, G., Raz, R., AND Safra, S. PCP characterizations of NP: Towards a polynomially-small error-probability. In Proc. 31th ACM Symp. on Theory of Computing (Atlanta, Georgia, 1–4 May 1999), pp. 29–40.Google Scholar
  8. 8.
    Friedl, K., AND Sudan, M. Some improvements to total degree tests. In Proc. 3rd Israel Symposium on Theoretical and Computing Systems (1995).Google Scholar
  9. 9.
    Håstad, J. Clique is hard to approximate within n1-ε. In Proc. 37nd IEEE Symp. on Foundations of Comp. Science (Burlington, Vermont, 14–16 Oct. 1996), pp. 627–636.Google Scholar
  10. 10.
    Håstad, J. Some optimal inapproximability results. In Proc. 29th ACM Symp. on Theory of Computing (El Paso, Texas, 4–6 May 1997), pp. 1–10.Google Scholar
  11. 11.
    Lund, C., Fortnow, L., Karloff, H., AND Nisan, N. Algebraic methods for interactive proof systems. In Proc. 31st IEEE Symp. on Foundations of Comp. Science (St. Louis, Missouri, 22–24 Oct. 1990), pp. 2–10.Google Scholar
  12. 12.
    Polishchuk, A., AND Spielman, D. A. Nearly-linear size holographic proofs. In Proc. 26th ACM Symp. on Theory of Computing (Montréal, Québec, Canada, 23–25 May 1994), pp. 194–203.Google Scholar
  13. 13.
    Raz, R., AND Safra, S. A sub-constant error-probability low-degree test, and a subconstant error-probability PCP characterization of NP. In Proc. 29th ACM Symp. on Theory of Computing (El Paso, Texas, 4–6 May 1997), pp. 475–484.Google Scholar
  14. 14.
    Rubinfeld, R., AND Sudan, M. Robust characterizations of polynomials with applications to program testing. SIAM Journal of Computing 25, 2 (Apr. 1996), 252–271.Google Scholar
  15. 15.
    Sudan, M. Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems. PhD thesis, University of California, Berkeley, Oct. 1992.Google Scholar
  16. 16.
    Szegedy, M. Many-valued logics and holographic proofs. In Automata, Languages and Programming, 26st International Colloquium (Prague, Czech Republic, 11-15 July 1999), J. Wiedermann, P. van Emde Boas, and M. Nielsen, Eds., vol. 1644 of Lecture Notes in Computer Science, Springer-Verlag, pp. 676–686.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Prahladh Harsha
    • 1
  • Madhu Sudan
    • 1
  1. 1.Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations