Approximating Boolean functions with depth-2 circuits

by Eric Blais and Li-Yang Tan

Oded's comments

This paper studies the DNF-complexity of approximating general and specific Boolean functions, where approximation means obtaining the value of the function of on an arbitrary large constant fraction of the domain. Interestingly, the picture that arises is different from the one for exactly computing these functions (see details in the beginning of the introduction). Although the DNF-complexity of functions is a rather ``weak'' complexity measure, it is cedrtainly a natural one, and it is quite odd that its systematic study was relatively neglected so far.

The abstract

We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function. In the first direction, our main positive results are the first non-trivial universal upper bounds on approximability by DNFs: Our techniques extend broadly to give strong universal upper bounds on approximability by various depth-2 circuits that generalize DNFs, including the intersection of halfspaces, low-degree PTFs, and unate functions. We show that the parameters of our constructions come close to matching the information-theoretic inapproximability of a random function. In the second direction our main positive result is the construction of an explicit DNF that approximates the parity function: Using Fourier analytic tools we show that our construction is essentially optimal not just within the class of DNFs, but also within the far more expressive classes of the intersection of halfspaces and intersection of unate functions.

See ECCC TR13-051/a>.

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