Relating Branching Program Size and Formula Size over the Full Binary Basis

  • Martin Sauerhoff
  • Ingo Wegener
  • Ralph Werchner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1563)


Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L*(f) over the restricted basis {∨, ∧, ⌝} is considered. It is well-known that C(f) ≤ 3 BP(f), BP(f)L*(f), L*(f)L(f) 2, and C(f)L(f) - 1. These estimates are optimal. But the inequality BP(f)L(f) 2 can be improved to BP(f) ≤ 1.360 L(f) β, where β = log4(3 + √5) < 1.195.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Babai, P. Pudlák, V. Rödl, and E. Szemerédi. Lower bounds to the complexity of symmetric Boolean functions. Theoretical Computer Science, 74:313–323, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. A. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences, 38:150–164, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Cai and R. J. Lipton. Subquadratic simulations of circuits by branching programs. In Proc. of the 30th IEEE Symp. on Foundations of Computer Science (FOCS), 568–573, 1989.Google Scholar
  4. 4.
    R. Cleve. Towards optimal simulations of formulas by bounded-width programs. Computational Complexity, 1:91–105, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete Mathematics. Addison-Wesley, 1994.Google Scholar
  6. 6.
    V. M. Krapchenko. Complexity of the realization of a linear function in the class of π-circuits. Math. Notes Acad. Sci. USSR, 10:21–23, 1971.CrossRefGoogle Scholar
  7. 7.
    É. I. Neciporuk. A Boolean function. Soviet Mathematics Doklady, 7(4):999–1000, 1966.Google Scholar
  8. 8.
    I. Wegener. The Complexity of Boolean Functions. Wiley-Teubner, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin Sauerhoff
    • 1
  • Ingo Wegener
    • 1
  • Ralph Werchner
    • 2
  1. 1.FB Informatik, LS 2Univ. DortmundDortmundGermany
  2. 2.FrankfurtGermany

Personalised recommendations