Skip to main content
SpringerLink
Log in
Book cover

Summer School on Theoretical Aspects of Computer Science

TACSci 2000: Theoretical Aspects of Computer Science pp 30–83Cite as

Exact and Approximate Testing/Correcting of Algebraic Functions: A Survey

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2292))

Abstract

In the late 80’s Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self-testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation of hardness of approximation results. Applications in areas like computer vision, machine learning, and self-correcting programs were also established.

In the self-testing problem one is interested in determining (maybe probabilistically) whether a function to which one has oracle access satisfies a given property. We consider the problem of testing algebraic functions and survey over a decade of research in the area. Special emphasis is given to illustrate the scenario where the problem takes place and to the main techniques used in the analysis of tests. A novel aspect of this work is the separation it advocates between the mathematical and algorithmic issues that arise in the theory of self-testing.

Gratefully acknowledges the support of Conicyt via Fondecyt No. 1981182 and Fondap in Applied Mathematics, 2000.

Partially supported by the EC thematic network RAND-APX IST-1999-14036. The participation at the Summer School was founded by the LRI (Orsay) and the IPM (Tehran).

Partially supported by the EC thematic network RAND-APX IST-1999-14036. The participation at the Summer School was founded by the EGIDE (Paris) and the IPM (Tehran).

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Ar, M. Blum, B. Codenotti, and P. Gemmell. Checking approximate computations over the reals. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 786–795, San Diego, California, May 1993. ACM.

    Google Scholar 

  2. N. Alon, S. Dar, M. Parnas, and D. Ron. Testing clustering. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science. IEEE, 2000. (To appear).

    Google Scholar 

  3. N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy. Efficient testing of large graphs. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 656–666, New York City, New York, October 1999. IEEE.

    Google Scholar 

  4. N. Alon, M. Krivelevich, I. Newman, and M. Szegedy. Regular languages are testable with a constant number of queries. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 645–655, New York City, New York, October 1999. IEEE.

    Google Scholar 

  5. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 14–23, Pittsburgh, Pennsylvania, October 1992. IEEE. Final version in [ALM+98].

    Google Scholar 

  6. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. J. of the Association for Computing Machinery, 45(3):505–555, 1998.

    MathSciNet  Google Scholar 

  7. N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., first edition, 1992.

    Google Scholar 

  8. S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 2–13, Pittsburgh, Pennsylvania, October 1992. IEEE.

    Google Scholar 

  9. S. Arora and M. Sudan. Improved low-degree testing and its applications. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 485–495, El Paso, Texas, May 1997. ACM.

    Google Scholar 

  10. M. Bellare, D. Coppersmith, J. Hoastad, M. Kiwi, and M. Sudan. Linearity testing in characteristic two. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 432–441, Milwaukee, Wisconsin, October 1995. IEEE.

    Google Scholar 

  11. L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pages 16–25, St. Louis, Missouri, October 1990. IEEE. Final version in [BFL91].

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. L. Babai, L. Fortnow, L. A. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 21–31, New Orleans, Louisiana, May 1991. ACM.

    Google Scholar 

  14. M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 294–304, San Diego, California, May 1993. ACM.

    Google Scholar 

  15. M. Blum and S. Kannan. Designing programs that check their work. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 86–97, Seattle, Washington, May 1989. ACM. Final version in [BK95].

    Google Scholar 

  16. M. Blum and S. Kannan. Designing programs that check their work. J. of the Association for Computing Machinery, 42(1):269–291, 1995.

    MATH  Google Scholar 

  17. M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 73–83, Baltimore, Maryland, May 1990. ACM. Final version in [BLR93].

    Google Scholar 

  18. M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. J. of Computer and System Sciences, 47(3):549–595, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Blum. Designing programs to check their work. Technical Report TR-88-009, International Computer Science Institure, 1988.

    Google Scholar 

  20. M. Bender and D. Ron. Testing acyclicity of directed graphs in sublinear time. In Proceedings of the 27th International Colloquium on Automata, Languages and Programming, volume 1853 of LNCS, pages 809–820. Springer-Verlag, 2000.

    Chapter  Google Scholar 

  21. M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 184–193, Montréal, Québec, Canada, May 1994. ACM.

    Google Scholar 

  22. M. Blum and H. Wasserman. Reflections on the Pentium division bug. IEEE Trans. Comp., 26(5):1411–1473, April 1997.

    Google Scholar 

  23. D. Coppersmith. Manuscript. Result described in [BLR90], December 1989.

    Google Scholar 

  24. Y. Dodis, O. Goldreich, E. Lehman, S. Rsakhodnikova, D. Ron, and A. Samorodnitsky. Improved testing algorithms for monotonicity. In Proceedings of RANDOM’99, volume 1671 of LNCS, pages 97–108. Springer-Verlag, 1999.

    Google Scholar 

  25. W. van Dam, F. Magniez, M. Mosca, and M. Santha. Self-testing of universal and fault-tolerant sets of quantum gates. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 688–696, Portland, Oregon, May 2000. ACM.

    Google Scholar 

  26. F. Ergün, S. Ravi Kumar, and R. Rubinfeld. Approximate checking of polynomials and functional equations. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 592–601, Burlington, Vermont, October 1996. IEEE.

    Google Scholar 

  27. F. Ergün. Testing multivariate linear functions: Overcoming the generator bottleneck. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, pages 407–416, Las Vegas, Nevada, May 1995. ACM.

    Google Scholar 

  28. F. Ergün, S. Sivakumar, and S. Ravi Kumar. Self-testing without the generator bottleneck. SIAM J. on Computing, 29(5):1630–1651, 2000.

    Article  MATH  Google Scholar 

  29. U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pages 2–12, San Juan, Puerto Rico, October 1991. IEEE.

    Google Scholar 

  30. G. L. Forti. Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae, 50:143–190, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  31. O. Goldreich, S. Goldwasser, E. Lehman, and D. Ron. Testing monotonicity. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pages 426–435, Palo Alto, California, November 1998. IEEE.

    Google Scholar 

  32. O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 339–348, Burlington, Vermont, October 1996. IEEE.

    Google Scholar 

  33. P. Gemmell, R. Lipton, R. Rubinfeld, M. Sudan, and A. Wigderson. Self-testing/correcting for polynomials and for approximate functions. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 32–42, New Orleans, Louisiana, May 1991. ACM.

    Google Scholar 

  34. O. Goldreich. Combinatorial property testing — A survey, volume 43 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 45–60. ACM/AMS, 1998.

    MathSciNet  Google Scholar 

  35. O. Goldreich. Talk given at the DIMACS Workshop on Sublinear Algorithms, September 2000.

    Google Scholar 

  36. O. Goldreich and D. Ron. Property testing in bounded degree graphs. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 406–415, El Paso, Texas, May 1997. ACM.

    Google Scholar 

  37. O. Goldreich and D. Ron. A sublinear bipartiteness tester for bounded degree graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 289–298, Dallas, Texas, May 1998. ACM.

    Google Scholar 

  38. O. Goldreich and D. Ron. On testing expansion in bounded-degree graphs. Technical Report ECCC TR00-020, Electronic Colloquium on Computational Complexity, 2000. (Available at http://www.eccc.uni-trier.de/eccc/).

  39. J. Håstad. Testing of the long code and hardness of clique. In Proceedings of the 37nd Annual IEEE Symposium on Foundations of Computer Science, pages 11–19, Burlington, Vermont, October 1996. IEEE.

    Google Scholar 

  40. J. Håstad. Getting optimal in-approximability results. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages 1–10, El Paso, Texas, May 1997. ACM.

    Google Scholar 

  41. D. H. Hyers and T. M. Rassias. Approximate homomorphisms. Aequationes Mathematicae, 44:125–153, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  42. D. H. Hyers. On the stability of the linear functional equation. Proceedings of the National Academy of Science, U.S.A., 27:222–224, 1941.

    Article  MATH  MathSciNet  Google Scholar 

  43. M. Kiwi. Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes. PhD thesis, Massachusetts Institute of Technology, February 1996.

    Google Scholar 

  44. M. Kiwi, F. Magniez, and M. Santha. Approximate testing with relative error. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages 51–60, Atlanta, Georgia, May 1999. ACM.

    Google Scholar 

  45. M. Kiwi and A. Russell. Linearity testing over prime fields. Unpublished manuscript, 1997.

    Google Scholar 

  46. R. J. Lipton. New directions in testing, volume 2 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 191–202. ACM/AMS, 1991.

    MathSciNet  Google Scholar 

  47. F. Magniez. Auto-test pour les calculs approché et quantique. PhD thesis, Université Paris-Sud, France, 2000.

    Google Scholar 

  48. F. Magniez. Multi-linearity self-testing with relative error. In Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, volume 1770 of LNCS, pages 302–313. Springer-Verlag, 2000.

    Google Scholar 

  49. M. Mishra, D. Oblinger, and L. Pirtt. Way-sublinear time approximate (PAC) clustering. Unpublished, 2000.

    Google Scholar 

  50. I. Newman. Testing of functions that have small width branching programs. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science. IEEE, 2000. (To appear).

    Google Scholar 

  51. M. Parnas and D. Ron. Testing the diameter of graphs. In Proceedings of RANDOM’99, volume 1671 of LNCS, pages 85–96. Springer-Verlag, 1999.

    Google Scholar 

  52. A. Polishchuk and D. Spielman. Nearly-linear size holographic proofs. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 194–203, Montréal, Québec, Canada, May 1994. ACM.

    Google Scholar 

  53. D. Ron. Property testing (A tutorial), 2000. (Available at http://www.eng.tau.ac.il/?danar/papers.html). To appear in Handbook on Randomization.

  54. R. Rubinfeld. A mathematical theory of self-checking, self-testing and self-correcting programs. PhD thesis, University of California, Berkeley, 1990.

    Google Scholar 

  55. R. Rubinfeld. On the robustness of functional equations. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 288–299, Santa Fe, New Mexico, November 1994. IEEE. Final version in [Rub99].

    Google Scholar 

  56. R. Rubinfeld. On the robustness of functional equations. SIAM J. on Computing, 28(6):1972–1997, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  57. T. M. Rassias and P. Šemrl. On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society, 114(4):989–993, April 1992.

    Article  MATH  MathSciNet  Google Scholar 

  58. R. Rubinfeld and M. Sudan. Testing polynomial functions efficiently and over rational domains. In Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 23–32, Orlando, Florida, January 1992. ACM/SIAM. Final version in [RS96].

    Google Scholar 

  59. R. Rubinfeld and M. Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal of Computing, 25(2):252–271, April 1996.

    Article  MATH  MathSciNet  Google Scholar 

  60. F. Skopf. Sull’approssimazione delle applicazioni localmente β-additive.Atti della Accademia delle Sciencze di Torino, 117:377–389, 1983. (In Italian.).

    Google Scholar 

  61. L. Trevisan. Recycling queries in PCPs and in linearity tests. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 299–308, Dallas, Texas, May 1998. ACM.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. Ing. Matemática, U. Chile & Ctr. Modelamiento Matemático, UMR 2071, UChile-CNRS, Santiago, 170-3, Chile

    Marcos Kiwi

  2. CNRS-LRI, UMR 8623 Université Paris-Sud, 91405, Orsay, France

    Frédéric Magniez & Miklos Santha

Authors
  1. Marcos Kiwi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frédéric Magniez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miklos Santha
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

    Gholamreza B. Khosrovshahi

  2. Dept. of Mathematics & Computer Science, Drexel University, 19104, Philadelphia, PA, USA

    Ali Shokoufandeh

  3. Digital Fountain, Inc., 39151 Civic Center Drive, 94538, Fremont, CA, USA

    Amin Shokrollahi

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kiwi, M., Magniez, F., Santha, M. (2002). Exact and Approximate Testing/Correcting of Algebraic Functions: A Survey. In: Khosrovshahi, G.B., Shokoufandeh, A., Shokrollahi, A. (eds) Theoretical Aspects of Computer Science. TACSci 2000. Lecture Notes in Computer Science, vol 2292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45878-6_2

Download citation

  • DOI: https://doi.org/10.1007/3-540-45878-6_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43328-6

  • Online ISBN: 978-3-540-45878-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation