Smallest Formulas for Parity of 2k Variables Are Essentially Unique

  • Jun Tarui
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


For n = 2 k , we know that the size of a smallest AND/OR/ NOT formula computing the Boolean function Open image in new window is exactly n 2: For any n, it is at least n 2 by classical Khrapchenko’s bound, and for n = 2 k we easily obtain a formula of size n 2 by writing and recursively expanding We show that for n = 2 k the formula obtained above is an essentially unique one that computes Open image in new window with size n 2. In the equivalent framework of the Karchmer-Wigderson communication game, our result means that an optimal protocol for Parity of 2 k variables is essentially unique.


Boolean Function Formula Computing Formula Size Exact Complexity Dual Case 
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.
    Håstad, J.: The Shrinkage Exponent of De Morgan Formulae is 2. SIAM Journal on Computing 27(1), 48–64 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Khrapchenko, V.M.: A Method of Determining Lower Bounds for the Complexity of Π-Schemes. Mat.Zametski 10(1), 83–92 (1971) (in Russian); English translation in: Math. Notes 10(1), 474–479 (1971) zbMATHGoogle Scholar
  3. 3.
    Karchmer, M., Wigderson, A.: Monotone Circuits for Connectivity Require Super-Logarithmic Depth. SIAM J. Discrete Mathematics 3(2), 255–265 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Karchmer, M.: Communication Complexity: A New Approach to Circuit Depth. MIT Press, Cambridge (1989)Google Scholar
  5. 5.
    Boppana, R., Sipser, M.: The Complexity of Finite Functions. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science volume A, Algorithms and Complexity. MIT Press, Cambridge (1990)Google Scholar
  6. 6.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge Univ. Press, Cambridge (1997)zbMATHGoogle Scholar
  7. 7.
    Wegener, I.: The Complexity of Boolean Functions. Wiley, Chichester (1987) (on-line copy available at the web site of ECCC under Monographs)zbMATHGoogle Scholar
  8. 8.
    Arora, S., Barak, B.: Complexity Theory: A Modern Approach (to be published, 2008)Google Scholar
  9. 9.
    Zwick, U.: An Extension of Khrapchenko’s Theorem. Information Processing Letters 37, 215–217 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Koutsoupias, E.: Improvements on Khraphchenko’s Theorem. Information Processing Letters 116, 399–403 (1993)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Laplante, S., Lee, T., Szegedy, M.: The Quantum Adversary Method and Classical Formula Size Lower Bounds. Computational Complexity 15(2), 163–196 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Borodin, A.: Horner’s Rule is Uniquely Optimal. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 45–58. Academic Press, London (1971)Google Scholar
  13. 13.
    Knuth, D.: The Art of Computer Programming, 3rd edn. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jun Tarui
    • 1
  1. 1.University of Electro-Comm,ChofuTokyoJapan

Personalised recommendations