A New Rank Technique for Formula Size Lower Bounds

  • Troy Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


We introduce a new technique for proving formula size lower bounds based on matrix rank. A simple form of this technique gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an n 2 lower bound on the parity function. Applying our method to the parity function, we are able to give an exact expression for the formula size of parity: if n = 2 + k, where 0 ≤ k < 2, then the formula size of parity on n bits is exactly 2(2 + 3k) = n 2 + k2 − k 2. Such a bound cannot be proven by any of the lower bound techniques of Khrapchenko, Nečiporuk, Koutsoupias, or the quantum adversary method, which are limited by n 2.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Amb02]
    Ambainis, A.: Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences 64, 750–767 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  2. [Amb03]
    Ambainis, A.: Polynomial degree vs. quantum query complexity. In: Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, pp. 230–239. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  3. [BSS03]
    Barnum, H., Saks, M., Szegedy, M.: Quantum decision trees and semidefinite programming. In: Proceedings of the 18th IEEE Conference on Computational Complexity, pp. 179–193. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  4. [Hås98]
    Håstad, J.: The shrinkage exponent is 2. SIAM Journal on Computing 27, 48–64 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  5. [Khr71]
    Khrapchenko, V.M.: Complexity of the realization of a linear function in the case of Π-circuits. Math. Notes Acad. Sciences 9, 21–23 (1971)CrossRefzbMATHGoogle Scholar
  6. [KKN95]
    Karchmer, M., Kushilevitz, E., Nisan, N.: Fractional covers and communication complexity. SIAM Journal on Discrete Mathematics 8(1), 76–92 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  7. [KN97]
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  8. [Kou93]
    Koutsoupias, E.: Improvements on Khrapchenko’s theorem. Theoretical Computer Science 116(2), 399–403 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  9. [KW88]
    Karchmer, M., Wigderson, A.: Monotone connectivity circuits require super-logarithmic depth. In: Proceedings of the 20th ACM Symposium on the Theory of Computing, pp. 539–550. ACM Press, New York (1988)Google Scholar
  10. [LLS06]
    Laplante, S., Lee, T., Szegedy, M.: The quantum adversary method and classical formula size lower bounds. Computational Complexity 15, 163–196 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  11. [MS82]
    Melhorn, K., Schmidt, E.: Las Vegas is better than determinism in VLSI and distributed computing. In: Proceedings of the 14th ACM Symposium on the Theory of Computing, pp. 330–337. ACM Press, New York (1982)Google Scholar
  12. [Neč66]
    Nečiporuk, E.I.: A Boolean function. Soviet Mathematics–Doklady 7, 999–1000 (1966)Google Scholar
  13. [PPZ92]
    Paterson, M., Pippenger, N., Zwick, U.: Optimal carry save networks. In: Boolean function complexity. London Mathematical Society Lecture Note Series, vol. 169, pp. 174–201. Cambridge University Press, Cambridge (1992)Google Scholar
  14. [Rad97]
    Radhakrishnan, J.: Better lower bounds for monotone threshold formulas. Journal of Computer and System Sciences 54(2), 221–226 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  15. [Raz90]
    Razborov, A.: Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica 10(1), 81–93 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  16. [Raz92]
    Razborov, A.: On submodular complexity measures. In: Paterson, M. (ed.) Boolean function complexity. London Math. Soc. Lecture Notes Series, vol. 169, pp. 76–83. Cambridge University Press, Cambridge (1992)Google Scholar
  17. [Val84]
    Valiant, L.G.: Short monotone formulae for the majority function. Journal of Algorithms 5, 363–366 (1984)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Troy Lee
    • 1
  1. 1.LRI, Université Paris-Sud 

Personalised recommendations