Learning a Function of r Relevant Variables

  • Avrim Blum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2777)


This problem has been around for a while but is one of my favorites. I will state it here in three forms, discuss a number of known results (some easy and some more intricate), and finally end with small financial incentives for various kinds of partial progress. This problem appears in various guises in [2, 3, 10]. To begin we need the following standard definition: a boolean function f over {0,1} n has (at most) r relevant variables if there exist r indices i 1, ..., i r such that \(f(x) = g(x_{i_1}, \ldots, x_{i_r})\) for some boolean function g over {0,1} r . In other words, the value of f is determined by only a subset of r of its n input variables. For instance, the function \(f(x) = x_1\bar{x}_2 \vee x_2\bar{x}_5 \vee x_5\bar{x}_1\) has three relevant variables. The “class of boolean functions with r relevant variables” is the set of all such functions, over all possible g and sets {i 1, ..., i r }.


Boolean Function Relevant Variable Target Function Polynomial Time Algorithm Statistical Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blum, A., Furst, M., Jackson, J., Kearns, M., Mansour, Y., Rudich, S.: Weakly learning DNF and characterizing statistical query learning using fourier analysis. In: Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pp. 253–262 (May 1994)Google Scholar
  2. 2.
    Blum, A., Furst, M., Kearns, M., Lipton, D.: Cryptographic primitives based on hard learning problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 278–291. Springer, Heidelberg (1994)Google Scholar
  3. 3.
    Blum, A.: Relevant examples and relevant features: Thoughts from computational learning theory. In: AAAI 1994 Fall Symposium, Workshop on Relevance (1994)Google Scholar
  4. 4.
    Bshouty, N.H.: Exact learning via the monotone theory. In: Proceedings of the IEEE Symposium on Foundation of Computer Science, Palo Alto, CA, pp. 302–311 (1993)Google Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness i: Basic results. SIAM Journal on Computing 24(4), 873–921 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Jackson, J.: An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. In: Proceedings of the IEEE Symposium on Foundation of Computer Science (1994)Google Scholar
  8. 8.
    Kearns, M.: Efficient noise-tolerant learning from statistical queries. Journal of the ACM 45(6), 983–1006 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kalai, A., Mansour, Y.: Perosnal communication (2001)Google Scholar
  10. 10.
    Mossel, E., O’Donnell, R., Servedio, R.: Learning juntas. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Avrim Blum
    • 1
  1. 1.Carnegie Mellon University 

Personalised recommendations