Symmetric functions in NC¹

Question

A boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ is symmetric if $f(x)$ depends only on the number of $1$s in $x$. It is known that every boolean function is in $\mathrm{NC}^1$, i.e. there is a circuit of depth $O(\log n)$ computing it.

What is known about the constant inside the $O()$ notation? Specifically, can one construct, for every $c$, a symmetric function requiring a circuit of depth at least $c \log n$? Or there is some constant $c_0$ such that every symmetric function has a circuit of depth at most $c_0\log n$?

Answer 1

We can treat the input $x$ as a boolean array. Using an $O(\log n)$ depth sorting network (such as the AKS network), we can sort $x$ in nonincreasing order. Calling the result $y$, using $O(1)$ more depth we can compute $z_i = y_i \land \lnot y_{i+1}$ (extended in the right way to the boundary). The vector $z_0,\ldots,z_n$ is the indicator vector of the Hamming weight of $x$. At this point you can compute any symmetric function using $O(\log n)$ more depth.

This shows that there is a universal constant $C$ such that all symmetric functions can be computed in depth $C\log n$ using constant fan-in circuits.

Answer 2

Fix a circuit of depth $c\log n$ computing the majority function $\def\M{\mathrm{Maj}}\M_{2n+1}\colon\{0,1\}^{2n+1}\to\{0,1\}$. (For example, the construction of Valiant gives $c<5{.}3$.) Then for any $k\le n+1$, the threshold function $$\theta^n_k(x_0,\dots,x_{n-1})=\begin{cases}1&\bigl|\{i<n:x_i=1\}\bigr|\ge k\\0&\text{otherwise}\end{cases}$$ can be defined by the circuit $$\M_{2n+1}(x_0,\dots,x_{n-1},\underbrace{0,\dots,0}_{k},\underbrace{1,\dots,1}_{n+1-k})$$ of depth $c\log n$, and an arbitrary symmetric function $f\colon\{0,1\}^n\to\{0,1\}$ can be written as $$f(\vec x)=\bigvee_{k\in I}\bigl(\theta^n_k(\vec x)\land\neg\theta^n_{k+1}(\vec x)\bigr)$$ for some $I\subseteq\{0,\dots,n\}$. Arranging the large $\bigvee$ as a balanced binary tree of disjunctions, this gives a circuit of depth at most $(c+1)\log n+2$. (E.g., using Valiant’s construction, every symmetric function has a circuit of depth $<6{.}3\log n$.)

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Concerning lower bounds, the only applicable lower bound I am aware of is that formulas over the De Morgan basis $\{\land,\lor,\neg,0,1\}$ computing Parity need size $\Omega(n^2)$, which implies that any De Morgan circuit for Parity has depth $\ge2\log n$ up to an additive constant. (This bound is tight for Parity, but presumably not for all symmetric functions.)