What if Boolean circuits has bounded depth?

Depth of Boolean circuit: the length of a longest path from an input gate to an output gate.

It’s a model of parallel computation where depth corresponds to parallel time.

Parallel model of random access machine PRAM ~~痛苦的计算几何课的回忆~~

📖Definition of $SIZE-DEPTH$ . Let $S:\mathbb N\to\mathbb N$ and $d:\mathbb N\to\mathbb N$ . $SIZE-DEPTH(S(n),d(n))$ is the class of languages computed by families of Boolean circuits of size $O(S(n))$ and depth $O(d(n))$ .

📖Definition of $NC^i$ . For $i\ge0$ , define $NC^i=SIZE-DEPTH(n^O(1),\log^i(n))$ , and further define $NC=\cup_{i\ge1}NC^i$ .

These circuit classes is restrict to polynomial size and poly-logarithmic depth.

By definition, $NC\subseteq P/poly$ , hence $U_L-NC\subseteq P$ .

为什么呢？回顾定义 $P/poly=\bigcup_{k\gt0}SIZE(n^k)$ ，发现它的多了一个深度的限制。

In fact $P\not\subseteq NC$ is expected to hold. This means not all languages in $P$ can benefit greatly from parallelism.

General Boolean Circuits

Consider Boolean circuits that allows arbitrary fanin gate and more general functions than $AND, OR, NOT$ .

General definition of Boolean circuit. A Boolean circuit $C$ with $n$ inputs and $m$ outputs consists of the following:

An underlying directed acyclic graph $D=(V,A)$ (multiple arcs is allowed)
A labeling $\ell$ of the nodes of $V$ .
A linear ordering on the arcs $A$ of $V$ .
A choice of $m$ output gates $o_1,\cdots,o_m\in V$

The size of $C$ is the number of wires. The labelling function $\ell$ should satisfy the following for all gates $g$ :

If the fanin of $g$ is $0$ , then $\ell(g)\in\{0,1,x_1,\cdots,x_n,\overline{x_1},\cdots,\overline{x_n}\}$ .
If the fanin of $g$ is $k\ge1$ , then $\ell(g)=f_g$ where $f_g:\{0,1\}^k\to\{0,1\}$ is an $k$ -ary Boolean function.

$C$ compute a Boolean function $f:\{0,1\}^n\to\{0,1\}^m$ in a natural way.

Boolean functions with k-ary

AND(z_1,\cdots,z_k)=z_1\and\cdots\and z_k\\ OR(z_1,\cdots,z_k)=z_1\or\cdots\or z_k\\ MOD_m(z_1,\cdots,z_k)=\begin{cases} 1\ if\ \sum_{i=1}^kz_i\not\equiv0\mod m\\ 0\ if\ \sum_{i=1}^kz_i\equiv0\mod m \end{cases}

$MOD_2$ is the same as the xor-sum.

The major voting function $MAJ$ :

MAJ(z_1,\cdots,z_k)=\begin{cases} 1\ if\ \sum_{i=1}^kz_i\ge\frac{k}{2}\\ 0\ if\ \sum_{i=1}^kz_i\lt\frac{k}{2} \end{cases}

Let $w=(w_1,\cdots,w_k)\in\mathbb R^k$ be weights and $t\in\mathbb R$ be a threshold. Then $THR_{w,t}$ is defined:

THR_{w,t}(z_1,\cdots,z_k)=\begin{cases} 1\ if\ \sum_{i=1}^kw_iz_i\ge t\\ 0\ if\ \sum_{i=1}^kw_iz_i\lt t \end{cases}

When all weights is $1$ ,

T_t(z_1,\cdots,z_k)=\begin{cases} 1\ if\ \sum_{i=1}^kz_i\ge t\\ 0\ if\ \sum_{i=1}^kz_i\lt t \end{cases}

Several Circuit Classes

📖Definition of $AC^i$ . For $i\ge0$ , $AC^i$ is the class of languages computed by families of Boolean circuits of polynomial size and depth $O(\log^in)$ using unbounded fanin $AND$ and $OR$ gates.

$AC^i$ is similar to $NC^i$ .

🤔Proposition 5. Hierarchy relation between $NC$ and $AC$ . $NC^i\subseteq AC^i\subseteq NC^{i+1}$ for all $i\ge0$ .

$NC^i\subseteq AC^i$ is obtained by definition.
For $AC^i\subseteq NC^{i+1}$ : $AND$ and $OR$ gates of fanin $k$ can be replaced by a binary tree of depth $\lceil\log_2k\rceil$ of corresponding fanin $2$ gates. Thus the depth increases by a factor $O(\log n)$ .

📖Definition of $AC^0,ACC^0$ . Let $m\gt1$ . $AC^0[m]$ is the class of languages computed by families of Boolean circuits of polynomial size and depth $O(1)$ using unbounded fanin $AND,OR,MOD_m$ gates. Define $ACC^0=\bigcup_{m\gt1}AC^0[m]$ .

📖Definition of $TC^0$ . $TC^0$ is the class of languages computed by families of Boolean circuits of polynomial size and depth $O(1)$ using unbounded fanin $MAJ$ gates only.

语言类 $TC^0$ 可以被看做神经网络。

Logarithmic Space and Logarithmic Depth Circuits

log-space can be roughly the same as log-depth circuits.

🎯Theorem 38. $U_L-NC^1\subseteq L$ .

Circuit $C$ with depth $(\log n)$ is computable by a $O(\log n)$ space Turing Machine.

📖Definition of Boolean product of matrices. $A=(a_{ij})\in\{0,1\}^{m\times r},B=(b_{ij})\in\{0,1\}^{r\times n}$ . The Boolean product $AB$ is the $m\times n$ Boolean matrix $C=(c_{ij})$ s.t.

c_{ij}=\bigvee_{k=1}^ra_{ik}\and a_{kj}.

A $n\times n$ Boolean matrix corresponds to a directed graph. The transitive closure of the graph gives rise to the transitive closure of the matrix.

回想起大二学的离散数学了吗？邻接矩阵的闭包与可达性。

📖Definition of transitive closure. Transitive closure $A^*$ of $A$ is the $n\times n$ Boolean matrix given by $A^*=\bigvee_{i\ge0}A^i$ , where $A^0$ is the identity matrix $I$ .

🎯Theorem 39. $NL\subseteq U_L-AC^1$

~~懒得看证明了，到学的时候再来~~

🤔Corollary 6. $U_L-NC^1\subseteq L\subseteq NL\subseteq U_L-AC^1$ .

其实这里也存在层次结构： $NL\subseteq U_L-AC^1\subseteq U_L-NC^2\subseteq L^2(=DSPACE(\log^2n))$ .

Functions and Reductions

Boolean circuit is fixed number of inputs, we consider output is also a bounded function of the length of input.

📖Definition of length-respecting. A function $f:\{0,1\}^*\to\{0,1\}^*$ is length-respecting if for all $x,y\in\{0,1\}^*$ , we have $|x|=|y|\Rarr|f(x)|=|f(y)|$ .

📖Definition of $AC^0$ Turing reduction. Let $f$ and $g$ be length-respecting Boolean functions. We say $f$ reduces to $g$ by an $AC^0$ Turing reduction if $f$ is computed by a family of Boolean functions of polynomial size and depth $O(1)$ using unbounded fanin $AND$ and $OR$ gates as well as gates computing any coordinate function of $g$ .

~~什么玩意啊，绕来绕去的~~

If $f$ reduces to $g$ by an $AC^0$ Turing reduction, it is denoted by $f\le_T^{AC^0}g$ .

Easy to see that $AC^0$ Turing reductions satisfy transitivity.

🤔Lemma 8. If $f\le_T^{AC^0}g$ and $g\le_T^{AC^0}h$ , then $f\le_T^{AC^0}h$ .

Addition and Multiplication

$ADD$ : given $n$ -bit numbers $x$ and $y$ , output $n+1$ -bit number $z$ s.t. $z=x+y$ .

$MULT$ : given $n$ -bit numbers $x$ and $y$ , output $2n$ -bit number $z$ s.t. $z=xy$ .

🎯Theorem 40. $ADD\in AC^0$ .

!!!

A generalization of $ADD$ : iterated addition of $n$ numbers each of $n$ bits.

$ITADD$ : given $n$ numbers $x^1,\cdots,x^n$ of $n$ bits each, output $n+\lceil\log_2n\rceil$ -bit number $z$ s.t. $z=x^1+\cdots+x^n$ .

🎯Theorem 41. $ITADD\in NC^1$ .

We can to reduce the task to the task of add $2$ numbers each of $n+O(\log n)$ bits. These may be added in depth $O(\log n)$ by converting $AC^0$ for $ADD$ to a $NC^1$ circuit.

🤔Corollary 7. $TC^0\subseteq NC^1$ .

Note that $MAJ\le_T^{AC^0}ITADD$ by computing the sum of the input bits and comparing to the threshold value.

🤔Proposition 6. $MULT\le_T^{AC^0}ITADD$ .

Given $n$ -bit numbers $x$ and $y$ , we form the $n$ numbers $z^i$ , each of $2n-1$ bits by letting $z^i=x_i2^iy$ for $0\le i\lt n$ . Then the product is simply the sum of the numbers $z^0,\cdots,z^{n-1}$ .

$BCOUNT$ : given $n$ -bit number $x$ , output $\lceil\log_2^n+1\rceil$ -bit number $z$ , the sum of each bit.

Clearly $BCOUNT\le_T^{AC^0}ITADD$ .

🤔Proposition 7. $ITADD\le_T^{AC^0}BCOUNT$

~~看不懂这个特例，但是图灵归约一般不是对称的~~

🤔Proposition 8. Let $f$ be a symmetric Boolean function. Then $f\in TC^0$ .

🤔Corollary 8. $ACC^0\subseteq TC^0$ .

🤔Proposition 9. $MAJ\le_T^{AC^0}MULT$ .

😭这些证明都不想看

🎯Theorem 42. $MULT$ is complete for $TC^0$ under $\le_T^{AC^0}$ reductions.

Linear Threshold Functions

Using integers of bit-size $O(n\log n)$ as weights is sufficient to realize any linear threshold function.

🤔Lemma 9. Let $f$ be a linear threshold function on $n$ bits. Then there exists integers $w_1,\cdots,w_n$ and $t$ of magnitude at most $(n+1)^{\frac{n+1}{2}}$ s.t. $f=THR_{w,t}$ .