computational complexity

, Volume 8, Issue 4, pp 357–370 | Cite as

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

  • S. Jukna
  • A. Razborov
  • P. Savicky
  • I. Wegener

Abstract.

It is known that if a Boolean function f in n variables has a DNF and a CNF of size \( \le N \) then f also has a (deterministic) decision tree of size exp(O(log n log2 N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp\( (\Omega({\rm log^2} N)) \) where N is the total number of monomials in minimal DNFs for f and ¬f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC0.

Keywords. Computational complexity, Boolean functions, decision trees, branching programs, P versus NP $ \cap $co-NP. 

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • S. Jukna
    • 1
  • A. Razborov
    • 2
  • P. Savicky
    • 3
  • I. Wegener
    • 4
  1. 1.Department of Computer Science, University of Trier, D-54286 Trier, Germany, and Institute of Mathematics, LT-2600 Vilnius, Lithuania, e-mail: jukna@ti.uni-trier.de DE
  2. 2.Steklov Mathematical Institute, Gubkina 8, 117966, Moscow, Russia, e-mail: razborov@genesis.mi.ras.ru RU
  3. 3.Institute of Computer Science, Academy of Sciences of Czech Republic, Pod vodárenskou vezi 2, 182 07 Praha 8, Czech Republic, e-mail: savicky@uivt.cas.cz CZ
  4. 4.Department of Computer Science, University of Dortmund, D-44221 Dortmund, Germany, e-mail: wegener@ls2.cs.uni-dortmund.de DE

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