Testing Formula Satisfaction

  • Eldar Fischer
  • Yonatan Goldhirsh
  • Oded Lachish
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

We study the query complexity of testing for properties defined by read once formulae, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in ε and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from satisfying the property. For formulae only involving And/Or gates, we provide a more efficient test whose query complexity is only quasipolynomial in ε. On the other hand we show that such testability results do not hold in general for formulae over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over alphabets of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Krivelevich, M., Newman, I., Szegedy, M.: Regular languages are testable with a constant number of queries. SIAM J. Comput. 30(6), 1842–1862 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ben-Sasson, E., Harsha, P., Lachish, O., Matsliah, A.: Sound 3-query pcpps are long. ACM Trans. Comput. Theory 1, 7:1–7:49 (2009)Google Scholar
  3. 3.
    Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SIAM J. Comput. 35(1), 1–21 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chakraborty, S., Fischer, E., Lachish, O., Matsliah, A., Newman, I.: Testing st-Connectivity. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 380–394. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Fischer, E.: The art of uninformed decisions: A primer to property testing. Current Trends in Theoretical Computer Science: The Challenge of the New Century I, 229–264 (2004)Google Scholar
  7. 7.
    Fischer, E., Goldhirsh, Y., Lachish, O.: Testing formula satisfaction, arXiv:1204.3413v1 [cs.DS]Google Scholar
  8. 8.
    Fischer, E., Lachish, O., Newman, I., Matsliah, A., Yahalom, O.: On the query complexity of testing orientations for being eulerian. TALG (to appear)Google Scholar
  9. 9.
    Fischer, E., Newman, I., Sgall, J.: Functions that have read-twice constant width branching programs are not necessarily testable. Random Struct. Algorithms 24(2), 175–193 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Fischer, E., Yahalom, O.: Testing convexity properties of tree colorings. Algorithmica 60(4), 766–805 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Goldreich, O.: A Brief Introduction to Property Testing. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 1–5. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45, 653–750 (1998)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Halevy, S., Lachish, O., Newman, I., Tsur, D.: Testing orientation properties. ECCC (153) (2005)Google Scholar
  14. 14.
    Halevy, S., Lachish, O., Newman, I., Tsur, D.: Testing properties of constraint-graphs. In: IEEE Conference on Computational Complexity (2007)Google Scholar
  15. 15.
    Newman, I.: Testing membership in languages that have small width branching programs. SIAM J. Comput. 31(5), 1557–1570 (2002)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Newman, I.: Property Testing of Massively Parametrized Problems - A survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 142–157. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Ron, D.: Property testing: A learning theory perspective. Found. Trends Mach. Learn. 1, 307–402 (2008)CrossRefGoogle Scholar
  18. 18.
    Ron, D.: Algorithmic and Analysis Techniques in Property Testing (2010)Google Scholar
  19. 19.
    Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eldar Fischer
    • 1
  • Yonatan Goldhirsh
    • 1
  • Oded Lachish
    • 2
  1. 1.Department of Computer ScienceTechnionHaifaIsrael
  2. 2.Birkbeck, University of LondonLondonUK

Personalised recommendations