On P versus NP
$ \cap $ co-NP for decision trees and read-once branching programs |
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Authors: | S Jukna A Razborov P Savicky I Wegener |
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Affiliation: | (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) Steklov Mathematical Institute, Gubkina 8, 117966, Moscow, Russia, e-mail: razborov@genesis.mi.ras.ru , RU;(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) Department of Computer Science, University of Dortmund, D-44221 Dortmund, Germany, e-mail: wegener@ls2.cs.uni-dortmund.de , DE |
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Abstract: | It is known that if a Boolean function f in n variables has a DNF and a CNF of size 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 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.
Received: June 5 1997. |
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Keywords: | , Computational complexity, Boolean functions, decision trees, branching programs, P versus NP $ \cap $co-NP, |
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