Bounds on the Size of Small Depth Circuits for Approximating Majority
نویسنده
چکیده
In this paper, we show that for every constant 0 < ǫ < 1/2 and for every constant d ≥ 2, the minimum size of a depth d Boolean circuit that ǫ-approximates Majority function on n variables is exp(Θ(n)). The lower bound for every d ≥ 2 and the upper bound for d = 2 have been previously shown by O’Donnell and Wimmer [ICALP’07], and the contribution of this paper is to give a matching upper bound for d ≥ 3.
منابع مشابه
Separating Ac from Depth-2 Majority Circuits∗
We construct a function in AC that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n1/5)). This solves an open problem due to Krause and Pudlák (1997) and matches Allender’s classic result (1989) that AC can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This...
متن کاملError bounds in approximating n-time differentiable functions of self-adjoint operators in Hilbert spaces via a Taylor's type expansion
On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in Hilbert Spaces via a Taylor's type expansion are given.
متن کاملTop-Down Lower Bounds for Depth 3 Circuits
We present a top-down lower bound method for depth 3 AND-OR-NOT circuits which is simpler than the previous methods and in some cases gives better lower bounds. In particular we prove that depth 3 AND-OR-NOT circuits that compute PARITY resp. MAJORITY require size a t least 20.6 18.. fi resp. 2°.849...fi. This is the first simple proof of a strong lower bound by a top-down argument for non-mono...
متن کاملApproximate Degree and the Complexity of Depth Three Circuits
Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the measure, but there may exist Bool...
متن کاملLecture 11 : Circuit Lower
There are specific kinds of circuits for which lower bounds techniques were successfully developed. One is small-depth circuits, the other is monotone circuits. For constant-depth circuits with AND,OR,NOT gates, people proved that they cannot compute simple functions like PARITY [3, 1] or MAJORITY. For monotone circuits, Alexander A. Razborov proved that CLIQUE, an NP-complete problem, has expo...
متن کامل