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Short Locally Testable Codes and Proofs: A Survey in Two Parts

  • Chapter
Property Testing

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6390))

Abstract

We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs. Local testability refers to approximately testing large objects based on a very small number of probes, each retrieving a single bit in the representation of the object. This yields super-fast approximate-testing of the corresponding property (i.e., be a codeword or a valid proof). We also review the related concept of local decodable codes.

The survey consists of two independent (i.e., self-contained) parts that cover the same material at different levels of rigor and detail. Still, in spite of the repetitions, there may be a benefit in reading both parts.

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References

  1. Alon, N., Krivelevich, M., Kaufman, T., Litsyn, S., Ron, D.: Testing low-degree polynomials over GF(2). In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 188–199. Springer, Heidelberg (2003)

    Google Scholar 

  2. Ambainis, A.: An upper bound on the communication complexity of private information retrieval. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 401–407. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  3. Arora, S.: Probabilistic checking of proofs and the hardness of approximation problems. PhD thesis, UC Berkeley (1994)

    Google Scholar 

  4. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998); Preliminary Version in 33rd FOCS (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998); Preliminary Version in 33rd FOCS (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1(1), 3–40 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: Proc. 23rd ACM Symposium on the Theory of Computing, pp. 21–31 (May 1991)

    Google Scholar 

  8. Barak, B.: How to go beyond the black-box simulation barrier. In: Proc. 42nd IEEE Symposium on Foundations of Computer Science, pp. 106–115 (October 2001)

    Google Scholar 

  9. Beimel, A., Ishai, Y., Kushilevitz, E., Raymond, J.F.: Breaking the O(n 1/(2k − 1)) barrier for information-theoretic private information retrieval. In: Proc. 43rd IEEE Symposium on Foundations of Computer Science, pp. 261–270 (November 2002)

    Google Scholar 

  10. Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M., Sudan, M.: Linearity testing in characteristic two. In: Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, pp. 432–441 (1995)

    Google Scholar 

  11. Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and nonapproximability—towards tight results. SIAM Journal on Computing 27(3), 804–915 (1998); Preliminary Version in 36th FOCS (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximation. In: Proc. 25th ACM Symposium on the Theory of Computing, pp. 294–304 (May 1993)

    Google Scholar 

  13. Bellare, M., Sudan, M.: Improved non-approximability results. In: Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pp. 184–193 (1994)

    Google Scholar 

  14. Ben-Sasson, E., Goldreich, O., Sudan, M.: Bounds on 2-query codeword testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 216–227. Springer, Heidelberg (2003)

    Google Scholar 

  15. Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. In: Proc. 36th ACM Symposium on the Theory of Computing, pp. 1–10 (June 2004); See ECCC Technical Report TR04-021 (March 2004)

    Google Scholar 

  16. Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Short PCPs verifiable in polylogarithmic time. In: 20th IEEE Conference on Computational Complexity, pp. 120–134 (2005)

    Google Scholar 

  17. Ben-Sasson, E., Guruswami, V., Kaufman, T., Sudan, M., Viderman, M.: Locally testable codes require redundant testers. In: 24th IEEE Conference on Computational Complexity, pp. 52–61 (2009)

    Google Scholar 

  18. Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. In: Proc. 35th ACM Symposium on the Theory of Computing, pp. 345–354 (June 2003)

    Google Scholar 

  19. Ben-Sasson, E., Sudan, M.: Robust locally testable codes and products of codes. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 286–297. Springer, Heidelberg (2004); See ECCC TR04-046 (2004)

    Chapter  Google Scholar 

  20. Ben-Sasson, E., Sudan, M.: Short PCPs with polylog query complexity. SIAM Journal on Computing 38(2), 551–607 (2008); Preliminary Version in 37th STOC (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ben-Sasson, E., Sudan, M., Vadhan, S., Wigderson, A.: Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In: Proc. 35th ACM Symposium on the Theory of Computing, pp. 612–621 (June 2003)

    Google Scholar 

  22. Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. Journal of Computer and System Science 47(3), 549–595 (1993); Preliminary Version in 22nd STOC (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Buhrman, H., de Wolf, R.: On relaxed locally decodable codes (July 2004) (unpublished manuscript)

    Google Scholar 

  24. Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. In: Proc. 30th ACM Symposium on the Theory of Computing, pp. 209–218 (May 1998)

    Google Scholar 

  25. Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private Information Retrieval. Journal of the ACM 45(6), 965–982 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dinur, I.: The PCP theorem by gap amplification. Journal of the ACM 54(3), Art. 12 (2007); Extended abstract in 38th STOC (2006)

    Google Scholar 

  27. Dinur, I., Harsha, P.: Composition of low-error 2-query PCPs using decodable PCPs. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 65–104. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  28. Dinur, I., Reingold, O.: Assignment-testers: Towards a combinatorial proof of the PCP-Theorem. SIAM Journal on Computing 36(4), 975–1024 (2006); Extended abstract in 45th FOCS (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Efremenko, K.: 3-query locally decodable codes of subexponential length. In: 41st ACM Symposium on the Theory of Computing, pp. 39–44 (2009)

    Google Scholar 

  30. Ergün, F., Kumar, R., Rubinfeld, R.: Fast approximate PCPs. In: Proc. 31st ACM Symposium on the Theory of Computing, pp. 41–50 (May 1999)

    Google Scholar 

  31. Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996); Preliminary version in 32nd FOCS (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Forney, G.D.: Concatenated Codes. MIT Press, Cambridge (1966)

    MATH  Google Scholar 

  33. Fortnow, L., Rompel, J., Sipser, M.: On the power of multi-prover interactive protocols. Theoretical Computer Science 134(2), 545–557 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proc. 3rd Israel Symposium on Theoretical and Computing Systems, Tel Aviv, Israel, January 4-6, pp. 190–198 (1995)

    Google Scholar 

  35. Gemmell, P., Lipton, R., Rubinfeld, R., Sudan, M., Wigderson, A.: Self-testing/correcting for polynomials and for approximate functions. In: Proc. 23rd ACM Symposium on the Theory of Computing, pp. 32–42 (1991)

    Google Scholar 

  36. Goldreich, O.: Short locally testable codes and proofs (survey). ECCC Technical Report TR05-014 (January 2005)

    Google Scholar 

  37. Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  38. Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM 45(4), 653–750 (1998); Preliminary Version in 37th FOCS (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Goldreich, O., Ron, D.: On proximity oblivious testing. ECCC, TR08-041 (2008); Also in the proceedings of the 41st STOC (2009)

    Google Scholar 

  40. Goldreich, O., Karloff, H., Schulman, L., Trevisan, L.: Lower bounds for linear locally decodable codes and private information retrieval. In: Proc. 17th Conference on Computational Complexity, Montréal, Québec, Canada, May 21-24, pp. 175–183 (2002)

    Google Scholar 

  41. Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almost linear length. In: Proc. 43rd IEEE Symposium on Foundations of Computer Science, pp. 13–22 (November 2002); See ECCC Report TR02-050 (2002)

    Google Scholar 

  42. Harsha, P., Sudan, M.: Small PCPs with low query complexity. Computational Complexity 9(3-4), 157–201 (2000); Preliminary Version in 18th STACS (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999); Preliminary Versions in 28th STOC (1996), and 37th FOCS (1997)

    Article  MathSciNet  Google Scholar 

  44. Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001); Preliminary Version in 29th STOC (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Katz, J., Trevisan, L.: On the efficiency of local decoding procedures for error-correcting codes. In: Proc. 32nd ACM Symposium on the Theory of Computing, pp. 80–86 (2000)

    Google Scholar 

  46. Kaufman, T., Litsyn, S., Xie, N.: Breaking the ε-soundness bound of the linearity test over GF(2). SIAM Journal on Computing 39(5), 1988–2003 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes via a quantum argument. In: Proc. 35th ACM Symposium on the Theory of Computing, pp. 106–115 (June 2003)

    Google Scholar 

  48. Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: Proc. 24th ACM Symposium on the Theory of Computing, pp. 723–732 (May 1992)

    Google Scholar 

  49. Lapidot, D., Shamir, A.: Fully parallelized multi prover protocols for NEXP-time. In: Proc. 32nd IEEE Symposium on Foundations of Computer Science, pp. 13–18 (October 1991) (extended abstract)

    Google Scholar 

  50. Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. Journal of the ACM 39(4), 859–868 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Meir, O.: Combinatorial construction of locally testable codes. SIAM Journal on Computing 39(2), 491–544 (2009); Extended abstrat in 40th STOC (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Meir, O.: Combinatorial PCPs with efficient verifiers. In: 50th IEEE Symposium on Foundations of Computer Science, pp. 463–471 (2009)

    Google Scholar 

  53. Micali, S.: Computationally sound proofs. SIAM Journal on Computing 30(4), 1253–1298 (2000); Preliminary Version in 35th FOCS (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. Moshkovitz, D., Raz, R.: Two query PCP with sub-constant error. In: 49th IEEE Symposium on Foundations of Computer Science, pp. 314–323 (2008)

    Google Scholar 

  55. Polishchuk, A., Spielman, D.A.: Nearly-linear size holographic proofs. In: Proc. 26th ACM Symposium on the Theory of Computing, pp. 194–203 (May 1994)

    Google Scholar 

  56. Raz, R.: A parallel repetition theorem. SIAM Journal of Computing 27(3), 763–803 (1998); Preliminary Version in 27th STOC (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996); Preliminary Version in 3rd SODA (1992)

    Article  MathSciNet  MATH  Google Scholar 

  58. Spielman, D.: Computationally efficient error-correcting codes and holographic proofs. PhD thesis, Massachusetts Institute of Technology (June 1995)

    Google Scholar 

  59. Sudan, M.: Efficient checking of polynomials and proofs and the hardness of approximation problems. Ph.D. Thesis, Computer Science Division, University of California at Berkeley (1992); Also appears as Lecture Notes in Computer Science, Vol. 1001, Springer (1996)

    Google Scholar 

  60. 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)

    Chapter  Google Scholar 

  61. Yekhanin, S.: Towards 3-Query locally decodable codes of subexponential length. In: 39th ACM Symposium on the Theory of Computing, pp. 266–274 (2007)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

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  1. Oded Goldreich
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  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 76100, Rehovot, Israel

    Oded Goldreich

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Goldreich, O. (2010). Short Locally Testable Codes and Proofs: A Survey in Two Parts. In: Goldreich, O. (eds) Property Testing. Lecture Notes in Computer Science, vol 6390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16367-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-16367-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16366-1

  • Online ISBN: 978-3-642-16367-8

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