Advertisement

Abstract

In a well-known result Goldreich and Trevisan (2003) showed that every testable graph property has a “canonical” tester in which a set of vertices is selected at random and the edges queried are the complete graph over the selected vertices. We define a similar-in-spirit canonical form for Boolean function testing algorithms, and show that under some mild conditions property testers for Boolean functions can be transformed into this canonical form.

Our first main result shows, roughly speaking, that every “nice” family of Boolean functions that has low noise sensitivity and is testable by an “independent tester,” has a canonical testing algorithm. Our second main result is similar but holds instead for families of Boolean functions that are closed under ID-negative minors. Taken together, these two results cover almost all of the constant-query Boolean function testing algorithms that we know of in the literature, and show that all of these testing algorithms can be automatically converted into a canonical form.

Keywords

property testing Boolean functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AKK+03]
    Alon, N., Kaufman, T., Krivelevich, M., 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. [Bla09]
    Blais, E.: Testing juntas nearly optimally. In: Proc. 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 151–158 (2009)Google Scholar
  3. [Bla10]
    Blais, E.: Personal communication (2010)Google Scholar
  4. [BLR93]
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comp. Sys. Sci. 47, 549–595 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [DLM+07]
    Diakonikolas, I., Lee, H., Matulef, K., Onak, K., Rubinfeld, R., Servedio, R., Wan, A.: Testing for concise representations. In: Proc. 48th Ann. Symposium on Computer Science (FOCS), pp. 549–558 (2007)Google Scholar
  6. [FKR+04]
    Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Computer & System Sciences 68(4), 753–787 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [GOS+09]
    Gopalan, P., O’Donnell, R., Servedio, R., Shpilka, A., Wimmer, K.: Testing Fourier dimensionality and sparsity. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 500–512. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. [GT03]
    Goldreich, O., Trevisan, L.: Three theorems regarding testing graph properties. Random Structures and Algorithms 23(1), 23–57 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [HR05]
    Hellerstein, L., Raghavan, V.: Exact learning of DNF formulas using DNF hypotheses. J. Comp. Syst. Sci. 70(4), 435–470 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [MORS10]
    Matulef, K., O’Donnell, R., Rubinfeld, R., Servedio, R.: Testing halfspaces. SIAM J. Comp. 39(5), 2004–2047 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [PRS02]
    Parnas, M., Ron, D., Samorodnitsky, A.: Testing Basic Boolean Formulae. SIAM J. Disc. Math. 16, 20–46 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dana Dachman-Soled
    • 1
  • Rocco A. Servedio
    • 1
  1. 1.Columbia UniversityNew YorkU.S.A.

Personalised recommendations