Formula Complexity of Ternary Majorities

  • Kenya Ueno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


It is known that any self-dual Boolean function can be decomposed into compositions of 3-bit majority functions. In this paper, we define a notion of a ternary majority formula, which is a ternary tree composed of nodes labeled by 3-bit majority functions and leaves labeled by literals. We study their complexity in terms of formula size. In particular, we prove upper and lower bounds for ternary majority formula size of several Boolean functions. To devise a general method to prove the ternary majority formula size lower bounds, we give an upper bound for the largest separation between ternary majority formula size and DeMorgan formula size.


Boolean Function Majority Function Parity Function Decomposition Scheme Full Adder 
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.
    Andreev, A.E.: On a method for obtaining more than quadratic effective lower bounds for the complexity of π-scheme. Moscow University Mathematics Bulletin 42(1), 63–66 (1987)zbMATHGoogle Scholar
  2. 2.
    Bioch, J.C., Ibaraki, T.: Decompositions of positive self-dual boolean functions. Discrete Mathematics 140(1-3), 23–46 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with boolean blocks, part I: Post’s lattice with applications to complexity theory. ACM SIGACT News 34(4), 38–52 (2003)CrossRefGoogle Scholar
  4. 4.
    Chockler, H., Zwick, U.: Which bases admit non-trivial shrinkage of formulae? Computational Complexity 10(1), 28–40 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fischer, M.J., Meyer, A.R., Paterson, M.S.: Ω(n log n) lower bounds on length of Boolean formulas. SIAM Journal on Computing 11(3), 416–427 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Håstad, J.: The shrinkage exponent of De Morgan formulas is 2. SIAM Journal on Computing 27(1), 48–64 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hrubeš, P., Jukna, S., Kulikov, A., Pudlák, P.: On convex complexity measures. Theoretical Computer Science 411, 1842–1854 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ibaraki, T., Kameda, T.: A theory of coteries: Mutual exclusion in distributed systems. IEEE Transactions on Parallel and Distributed Computing PDS-4(7), 779–794 (1993)CrossRefGoogle Scholar
  9. 9.
    Karchmer, M., Kushilevitz, E., Nisan, N.: Fractional covers and communication complexity. SIAM Journal on Discrete Mathematics 8(1), 76–92 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Khrapchenko, V.M.: Complexity of the realization of a linear function in the case of π-circuits. Mathematical Notes 9, 21–23 (1971)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lee, T.: A New Rank Technique for Formula Size Lower Bounds. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 145–156. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Neciporuk, E.I.: A boolean function. DOKLADY: Russian Academy of Sciences Doklady. Mathematics (formerly Soviet Mathematics–Doklady) 7, 999–1000 (1966)Google Scholar
  14. 14.
    Paterson, M.S., 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 (1992)Google Scholar
  15. 15.
    Paterson, M.S., Zwick, U.: Shallow circuits and concise formulae for multiple addition and multiplication. Computational Complexity 3(3), 262–291 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Post, E.L.: The two-valued iterative systems of mathematical logic. Annals Mathematical Studies, vol. 5. Princeton University Press (1941)Google Scholar
  17. 17.
    Pratt, V.R.: The effect of basis on size of Boolean expressions. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science (FOCS 1975), October 13-15, pp. 119–121. IEEE (1975)Google Scholar
  18. 18.
    Valiant, L.G.: Short monotone formulae for the majority function. Journal of Algorithms 5(3), 363–366 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kenya Ueno
    • 1
  1. 1.The Hakubi Center for Advanced Research and Graduate School of InformaticsKyoto UniversityJapan

Personalised recommendations