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\lecture{4}{February 14, 2005}{Madhu Sudan}{Elena Grigorescu}
\section{Overview}
In this lecture we will present Barrington's Theorem and the proof due to Ben-Or \& Cleve. We will also discuss some lower bounds on the depth of a circuit, and introduce the Switching Lemma.
\section{Review}
Recall the following definitions from the last lecture :
\BE
\item \textbf{Branching Program(BP)}:
\BI
\item DAG with 1 source, 2 sinks labeled $0/1$
\item the non-sink vertices are labeled by $\{x_1, \ldots, x_n\}$ and have 2 out-edges labeled $0/1$.
\EI
\item \textbf{Layered Branching Program(LBP)}:
\BI
\item the edges go only from one layer to the next
\item the width of a LBP is the max number of vertices in one layer
\EI
\EE
Notice that the notion of width of a LBP can be connected to that of space in a non-uniform setting.
Recall also that constant-width BP's can compute {\it{counting mod $k$}}, but it is not clear if, for example, the {\it majority function} can be computed by such BP's.
\section{Arithmetic Circuits and Register Machines}
An \textbf{arithmetic circuit} computes an arithmetic function of its inputs. Instead of gates computing boolean operations, we now have gates capable of addition and multiplication over some field of elements ( $F_2, F_5$, etc).
Similarly, one can define arithmetic formulas.
\noindent A \textbf{register machine } is a way of viewing a circuit as a program.
For example, the following code represents a register machine with an unbounded amount of space:
\begin{eqnarray*}
&G_1\leftarrow x_1+x_2\\
&G_2\leftarrow x_3*x_4\\
&G_3\leftarrow x_5-G_1\\
&\vdots\\
&G_m\leftarrow \ldots\\
&\vdots.
\end{eqnarray*}
We can restrict the amount of memory allowed, by defining registers $R_1, \ldots, R_l$ and writing the circuit in terms of the content of each register at each step.
Example:
\begin{eqnarray*}
&R_1\leftarrow x_1+x_2\\
&R_2\leftarrow x_3*x_4\\
&R_3\leftarrow x_5-R_1\\
&\vdots.
\end{eqnarray*}
%The size of the program is the number of instructions of the program.
\textbf{ Exercise:} Show that if all operations in an $l$-Register Machine are of the form $R_i\leftarrow R_j+x_l *R_k$ (over $F_2$), then the Register Machine can be simulated by a width $2^l~$ BP.
In the next section we address the converse problem, namely that of proving that depth $O(d)$ formulas can be converted into arithmetic formulas.
\section{Barrington's Theorem, Ben-Or \& Cleve's proof}
In the '80s, it was conjectured that the {\it majority of $n$ bits function} does not have $O(1)$-width and poly-size BP's. However, Barrington('86) showed that the conjecture was false.
\begin{theorem}[Barrington]
Any depth-$d$ formula over \{NOT, AND, OR\} has a BP of size $4^d$, and width $5.$
\end{theorem}
The proof of Barrington's theorem involves the notion of {\it permutation BP} and uses group-theoretic results. We will omit the original proof in this presentation, and instead show Ben-Or \& Cleve's proof of an equivalent result.
\begin{theorem}[Ben-Or, Cleve]
Any depth-$d$ arithmetic formula has a 3-register, $4^d$-size Register Machine with instructions of the form $R_i\leftarrow R_j+x_l*R_k$.
\end{theorem}
\begin{proof}
Inductive claim: If $f$ is a depth-$d$ formula, then there exists a $4^d$-size machine which starts with $R_0, R_1, R_2$ and computes $R_2+fR_1$.
Case: $f(x_1, x_2, \ldots, x_n)=x_1$
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Case: $f(x_1, x_2, \ldots, x_n)=f_1(x_1, \ldots, x_n)+f_2(x_1, \ldots, x_n)$, where $f_1, f_2$ have each depth $d-1$.
By induction we know $R_2+f_1R_1$ and $R_2+f_2R_1$. The following figure shows
the register assignments.
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Case: $f=f_1 f_2$ (trickier). The figure below shows how the computation proceeds after each $4^{d-1}$ steps. Along the way some registers are renamed. Note that the total number of steps of the computation is $4^d$.
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\end{proof}
\section{Concluding facts about BPs}
\BI
\item We could deduce a powerful upper bound
\item We do not know of significant lower bounds
\item Open Question: Are poly sized BP's the same as bounded width poly sized BP's?
\item Open Question: Is there an explicit function which does not have poly-size BP's?
\item Ajtai showed an explicit function that requires super linear depth (time), unless width/size $\geq 2^{\epsilon n}.$ In a future lecture we will present a version of this result in the uniform setting.
\EI
\section{Circuits}
In the case of circuits over \{2-OR, 2-AND, NOT\}-basis, there
exists an explicit construction of an $f$ with
circuit-size $\geq 4.5 n-o(n)~$[Raz, '02].
We are interested in two types of circuits:
\BI
\item Monotone circuits- those whose output does not decrease as an input changes from 0 to 1.(Eg: the circuit which, given the adjacency matrix of a graph, decides whether the graph has a click of size $k$).
\item $(\infty-OR, \infty-AND, NOT)$-circuits of bounded depth( AC$^0$-circuits)
(Note: in general AC$^i$ circuits have poly size and $(log n)^i$ depth.)
\EI
Known results regarding AC$^0$ circuits:
\BI
\item AC$^0$ circuits cannot compute the parity of $n$ bits.
\item Furst, Saxe, Sipser ('81) showed that the parity function requires circuits of super poly size.
\item Ajtai obtains independently the same results.
\item Yao ('85) obtains exponential lower bounds on size.
\item Hastad ('87) obtains similar results as Yao, but in a cleaner way, introducing the {\it Switching Lemma}.
\item Razborov-Smolensky prove similar results, using algebraic and probabilistic approaches.
\EI
Note: We will discuss the last two papers in future lectures.
\section{Switching Lemma}
Our goal is to prove that the parity function does not have a small depth circuit.
The idea is that there is a way to convert a DNF- formula( OR of AND's) into a CNF-formula (AND of OR's), and vice versa. However, in this conversion the sizes do not remain proportional. A way of controlling the sizes is by imposing some restrictions on the variables. For example, consider a random restriction obtained by tossing a 3-sided coin, assigning to each variable in a clause the value 0 w.p. ${{(1-p)\over{2}}}$, 1 w.p. ${{(1-p)\over{2}}}$, and leaving the variable unchanged w.p. p. In this way, we could reduce the depth of the circuit inductively.
We can now introduce the Switching Lemma.
Define a $k$-DNF($k$-CNF) to be a DNF(CNF) formula that contains at most $k$ literals in each clause.
\begin{lemma}[Switching Lemma]
If $F$ is an $l$- CNF formula on $n$ variables and we apply to it a random restriction with parameter $p\leq {1\over 2}$, then the resulting function has a $k$-DNF form with probability$\geq 1-(7pl)^k$.
\end{lemma}
We will show a proof of the lemma in the next lecture.
%\references
%\bibitem{1} Bounded-Width polynomial-Size Branching Programs recognize Exactly Those Languages in $NC^0$, David Barrington, ACM 1986
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