Optimal bounds on the approximation of boolean functions with consequences on the concept of hardness

Extended abstract
  • Alexander E. Andreev
  • Andrea E. F. Clementi
  • José D. P. Rolim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


We prove an optimal bound for the function L(n, m, ε) that gives the worst-case circuit-size complexity to approximate partial boolean functions having n inputs and domain size m within degree at least ε. Our bound applies to any partial boolean function and any approximation degree, completing the study of boolean function approximation introduced in [15]. We also provide the approximation degree (i.e. the value ε) achieved by polynomial size circuits on a ‘random’ boolean function. Our results give a new upper bound for the hardness function h(f), the function denoting the minimum value l for which there exists a circuit of size at most l that approximates a boolean function f with degree at least 1/l [14]. The contribution in the proof of the upper bound for L(n, m, ε) can be viewed as a set of technical results that globally show how boolean linear operators are “well” distributed over the class of 4-regular domains. We show how to apply this property to approximate partial boolean functions on general domains.


Linear Operator Boolean Function Domain Size General Domain Pseudorandom Generator 
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  1. 1.
    Allender E. and Strauss M. (1994), “Measure on Small Complexity Classes, with Applications for BPP”, Proc. of 35th IEEE FOCS, 819–830.Google Scholar
  2. 2.
    Alon N. and Spencer J.H. (1992), The Probabilistic Method, Wiley-Interscience Publication.Google Scholar
  3. 3.
    Andreev A.E. (1985), “The universal principle of self-correction”, Mat. Sbornik 127(169), N 6, 147–172 (in Russian). English transl. in Math. USSR Sbornik, 55 (1), 145–169 (1986).Google Scholar
  4. 4.
    Andreev A.E. (1989), “On the complexity of the realizations of partial boolean functions by circuits of functional elements”, J. of Discr. Math. and Appl. 1, 251–262 (1989).Google Scholar
  5. 5.
    Blum M. and Micali S. (1984), “How to generate cryptographically strong sequences of pseudorandom bits”, J. SIAM, 13(4), 850–864.CrossRefGoogle Scholar
  6. 6.
    Boppana R. and Hirshfield (1989), “Pseudorandom generators and complexity classes”, in Randomness and Computation (S. Micali Ed.), 5, 1–26, Adv. in Comput. Res., JAI Press.Google Scholar
  7. 7.
    Chernoff H. (1952), “A measure of asymptotic efficiency for tests of a hypothesis on the sum of observations”, Ann. Math. Statistic., 23, 493–507.Google Scholar
  8. 8.
    Feller W. (1970), An Introduction to probability theory and its applications, (Vol. I), III edition, John Wiley and Sons.Google Scholar
  9. 9.
    Hagerup T. and Rub C. (1990), “A guided tour of Chernoff bounds”, IPL, 33, 305–308.Google Scholar
  10. 10.
    Kearns M. and Vazirani U. (1994), Topics in Computational Learning Theory, MIT Press.Google Scholar
  11. 11.
    Lupanov O.B. (1965), “About a method circuits design — local coding principle”, Problemy Kibernet. 10, 31–110 (in Russian). English Translation in Systems Theory Res., 10, 1963.Google Scholar
  12. 12.
    Nechiporuk E.I. (1965), “About the complexity of gating circuits for the partial boolean matrix”, Dokl. Akad. Nauk SSSR, 163, 40–42 (in Russian). English translation in Soviet Math. Docl.Google Scholar
  13. 13.
    Nisan N. (1992), Using Hard Problems to Create Pseudorandom Generators, ACM Distinguished Dissertation, MIT Press.Google Scholar
  14. 14.
    Nisan N. and Wigderson A. (1994), “Hardness vs Randomness”, J. Comput. System Sci. 49, 149–167 (presented also at the 29th IEEE FOCS, 1988).Google Scholar
  15. 15.
    Pippenger N. (1977), “Information theory and the complexity of Boolean functions”, Math. Systems Theory 10, 129–167.CrossRefGoogle Scholar
  16. 16.
    Razborov A. and Rudich S. (1994), “Natural Proofs”, Proc. of 26th ACM STOC, 204–213, 1994.Google Scholar
  17. 17.
    Regan K.W., Sivakumar D. and Cai J. (1995), “Pseudorandom Generators, Measure Theory, and Natural Proofs”, Proc. of the 36th IEEE FOCS, to appear.Google Scholar
  18. 18.
    Shannon, C.E. (1949), “The synthesis of two-terminal switching circuits”, Bell. Syst. Tech. J. 28, 59–98.Google Scholar
  19. 19.
    Yao A.C. (1982), “Theory and applications of trapdoor functions”, Proc. of the 23th IEEE FOCS, 80–91.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Alexander E. Andreev
    • 1
  • Andrea E. F. Clementi
    • 2
  • José D. P. Rolim
    • 2
  1. 1.Department of MathematicsUniversity of Moscow, RUUSSR
  2. 2.Centre Universitaire d'InformatiqueUniversity of Geneva, CHSwitzerland

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