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.
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Received: June 5 1997.
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Jukna, S., Razborov, A., Savicky, P. et al. On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs. Comput. complex. 8, 357–370 (1999). https://doi.org/10.1007/s000370050005
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DOI: https://doi.org/10.1007/s000370050005