On DNF Approximators for Monotone Boolean Functions

  • Eric Blais
  • Johan Håstad
  • Rocco A. Servedio
  • Li-Yang Tan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be ε-approximated by a DNF g of size \(2^{n-\Omega_\epsilon(\sqrt{n})}\) satisfying g(x) ≤ f(x) for all x ∈ {0,1}n. This is the first non-trivial universal upper bound even for DNF approximators incurring two-sided error.

Next, we study the power of negations in DNF approximators for monotone functions. We exhibit monotone functions for which non-monotone DNFs perform better than monotone ones, giving separations with respect to both DNF size and width. Our results, when taken together with a classical theorem of Quine [1], highlight an interesting contrast between approximation and exact computation in the DNF complexity of monotone functions, and they add to a line of work on the surprising role of negations in monotone complexity [2,3,4].


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Eric Blais
    • 1
  • Johan Håstad
    • 2
  • Rocco A. Servedio
    • 3
  • Li-Yang Tan
    • 3
  1. 1.MITUSA
  2. 2.KTH Royal Institute of TechnologyUSA
  3. 3.Columbia UniversityUSA

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