Parity function is not in $AC^0$.

$MAJ\not\in AC^0$.

$NEXP\not\subseteq ACC^0$.

Some unknown proposition:

whether $NP\subseteq ACC^0$.
whether $EXP\subseteq ACC^0$.
whether $NEXP\subseteq TC^0$.
whether all languages in $NEXP$ can be computed by depth $2$ circuits consisting of weighted linear threshold gates.

The Razborov-Smolensky Lower Bound

The lower bounds for class $AC^0[p^r]$, where $p$ is a prime.

Main idea: convert Boolean circuit of small size into a low-degree probabilistic polynomial.

Here we consider $r=1$.

Definition 43. Let $\mathbb F$ be a field. By a probabilistic polynomial $P$ in $n$ variables of total degree $d$ over $\mathbb F$ we mean a random variable assuming as value such polynomials. Let $f:\{0,1\}^n\to\{0,1\}$ be a Boolean function. We say that $P$ computes $f$ with error $\epsilon$ if $\Pr[P(x)=f(x)]\ge1-\epsilon$ for all $x\in\{0,1\}^n$. Similarly we say that $P$ computes $f$ with one-sided error $\epsilon$ if $P$ computes $f$ with error $\epsilon$ and furthermore we have $\Pr[P(x)=f(x)]=1$ for all $x\in\{0,1\}^n$ where $f(x)=0$.

Then construct a probabilistic polynomial form an $AC^0[p]$ circuit.

Lemma 13. Let $p$ be a prime and let $C$ be a depth $h$ circuit with at most $S$ unbounded fanin $AND,OR,MOD_p$ as well as negation gates computing a Boolean function $f$. Let $t\ge1$ be an arbitrary integer. Then for any $n$ there exist a probabilistic polynomial in $n$ variables of total degree at most $(pt)^h$ over $\mathbb Z_p$ computing $f$ with error $\frac{S}{p^t}$.

Probabilistic polynomial for each of the gates of $C$. Each of these will be of degree at most $pt$ and compute the gate function with error $\frac{1}{p^t}$.

Composing these will give probabilistic polynomial of degree at most $(pt)^h$ computing $f$ and with error at most $\frac{S}{p^t}$ by a union bound.

Input gates: $x_i$ can be computed by the polynomials $x_i$ with error $0$. $\bar x_i$ can be computed by the polynomials $1-x_i$ with error $0$.

Negation gates: $\neg z$ can be computed by the polynomial $1-y$ with error $0$.

$MOD_p$ gate with inputs $z_1,\cdots,z_k$ can be computed by the polynomial

\[ \left(\sum_{i=1}^kz_i\right)^{p-1} \]

with error $0$ by Fermat's little theorem.

$OR$ gate with inputs $z_1,\cdots,z_k$: Consider picking $a_1,\cdots,a_k\in\mathbb Z_p$ uniformly and independently at random. Using Fermat's little theorem again, we have the polynomial

\[ \left(\sum_{i=1}^ka_iz_i\right)^{p-1} \]

computes $OR(z_1,\cdots,z_k)$ with one-sided error $\frac{1}{p}$. In order to decrease the error, use success amplification using $t$ independent trials.

\[ 1-\prod_{j=1}^t\left(1-\left(\sum_{i=1}^ka_{ij}z_i\right)^{p-1}\right), \]

which then computes $OR(z_1,\cdots,z_k)$ with one-sided error $\frac{1}{p^t}$.

Analogously, for $AND$ gate, the probabilistic polynomial

\[ 1-\prod_{j=1}^t\left(1-\left(\sum_{i=1}^ka_{ij}(1-z_i)\right)^{p-1}\right). \]

Lemma 14. Let $P$ be a probabilistic polynomial of degree $d$ and let $f:\{a,b\}^n\to\mathbb F$ be a function s.t. $\Pr[P(x)=f(x)]\ge1-\epsilon$. Then there exists a polynomial $P'$ of degree $d$ and a set $G\subseteq\{a,b\}^n$ with $|G|\ge(1-\epsilon)2^n$ s.t. $P(x)=f(x)$ for all $x\in G$.

Linearity of expectation:

\[ \mathbb E[|x\in\{a,b\}^n:P(x)=f(x)|]=\sum_{x\in\{a,b\}^n}\Pr[P(x)=f(x)]\ge2^n(1-\epsilon), \]

A low-degree polynomial is equal to the product of all variables on a set $G\subseteq\{a,b\}^n$ enables one to replace all polynomials by equivalent polynomials on the set $G$ of degree significantly smaller than $n$.

Lemma 15. Let $P$ be a polynomial in $n$ variables of total degree $\Delta$ over a field $\mathbb F$. Let $a,b\in\mathbb F$ and suppose $G\subseteq\{a,b\}^n$ is s.t. $$ \forall y\in G:P(y)=\prod_{i=1}^ny_i. $$ Then for every function $f:G\to\mathbb F$ there exists a polynomial $P_f$ of total degree at most $\frac{n+\Delta}{2}$ s.t. $$ \forall y\in G:P_f(y)=f(y). $$

Let $f:G\to\mathbb F$ be any function. Define multilinear polynomial $Q_f$: $$ Q_f(y)=\sum_{x\in G}\left(f(x)\left( \prod_{i:x_i=a}\frac{b-y_i}{b-a} \right)\left( \prod_{i:x_i=b}\frac{a-y_i}{a-b} \right)\right). $$ Clearly, $Q_f(y)=f(y)\forall y\in G$. Using $P$, any monomial of degree $d$ can be replaced by an equivalent polynomial of degree $n-d+\Delta$ since $$ \prod_{i\in I}y_i=\left(\prod_{i=1}^ny_i\right)\left(\prod_{i\not\in I}\frac{1}{y_i}\right)= P(y)\prod_{i\not\in I}\left(\frac{1}{a}\frac{b-y_i}{b-a}+\frac{1}{b}\frac{a-y_i}{a-b}\right). $$ Thus we replace each monomial of $Q_f$ of degree $d\gt\frac{n+\Delta}{2}$ by an equivalent polynomial of degree $n-d+\Delta\lt\frac{n+\Delta}{2}$. We can thus take $P_f$ to be the sum of these together with the monomials of $Q_f$ degree at most $\frac{n+\Delta}{2}$.

One may convert any polynomial into a multi-linear polynomial of the same degree which is equivalent on a given hypercube $\{a,b\}^n$.

Lemma 16. Let $a,b\in\mathbb F$ and let $P$ be a polynomial of degree $d$. Then there exists a multi-linear polynomial $P'$ of degree at most $d$ s.t. $P(x)=P'(x)$ for all $x\in\{a,b\}^n$.

An occurrence of $x_i^k$ in $P$ may be replaced by the polynomial

\[ \frac{x_i-b}{a-b}a^k+\frac{x_i-a}{b-a}b^k, \]

which can never increase the degree.

Lemma 17. For all integers $n\ge 1$:

\[ \frac{4^n}{\sqrt{\pi(n+1)}}\le C_{2n}^n\le\frac{4^n}{\sqrt{\pi n}} \]

Corollary 10. For all integers $n\ge1$:

\[ C_{\lceil n/2\rceil}^n\le\frac{2^n}{\sqrt{\pi\lceil n/2\rceil}}. \]

Proposition 13. Let $P$ be a polynomial in $n$ variables of total degree $\Delta$ over a field $\mathbb F$. Let $a,b\in\mathbb F$ and suppose $G\subseteq \{a,b\}^n$ is s.t.

\[ y\in G:P(y)=\prod_{i=1}^ny_i. \]

Then

\[ |G|\le 2^n\left(\frac{1}{2}+\frac{4}{10}\frac{\Delta}{\sqrt n}\right). \]

By Lemma 16, assume that $P$ is multilinear.

The number of functions $f:G\to\mathbb F$, which amounts to $p^{|G|}$, must be less than the number of multilinear polynomials of degree at most $\frac{n+\Delta}{2}$, which amounts to $p^M$, where $M$ denotes the number of monomials on $n$ variables of degree at most $\frac{n+\Delta}{2}$. It follows that $|G|\le M$.

Then conclude by estimating $M$.

\[ M=\sum_{i=0}^{\lfloor\frac{n+\Delta}{2}\rfloor}C_n^i=2^{n-1}+\sum_{i=\lceil n/2\rceil}^{\lfloor\frac{n+\Delta}{2}\rfloor}C_n^i\le2^{n-1}+\frac{\Delta}{2}\frac{2^n}{\sqrt{\pi n/2}}\le 2^{n-1}+\frac{4}{10}\frac{\Delta2^n}{\sqrt n}, \]

where $C_n^i$ is estimated by central binomial coefficient, for $i\gt n/2$, and use $\sqrt{2\pi}\ge\frac{10}{4}$,

Finally combine the construction of the probabilistic polynomials with the counting argument to derive a circuit lower bound.

Theorem 46 . Let $p\neq 2$ be a prime. Suppose that $C$ is depth $h$ circuit with at most $S$ unbounded fanin $AND,OR,MOD_p$ as well as $NEG$ gates computing $MOD_2$. Then

\[ S=2^{\Omega(n^{\frac{1}{2h}})}. \]

Lemma 13, Lemma 14, Proposition 13.

Theorem 47 (Smolensky). Let $p$ be a prime. Suppose that $C$ is a depth $h$ circuit with at most $S$ unbounded fanin $AND,OR,MOD_{p^r}$ gates for a constant $r\ge1$ as well as $NEG$ gates computing function $MOD_q$, where $q\neq p$ is a prime. Then

\[ S=2^{\Omega(n^{\frac{1}{2h}})}. \]

~~这个电路下界真的是人想的出来的吗！！！~~