Changes from 021221 meeting implemented.

circuits-model-runtime.tex

@@ -1,6 +1,7 @@
 %!TEX root=./main.tex
 \revision{
 \section{More on Circuits and Moments}
+
 }
 \label{sec:gen}
 %In this section, we consider generalizations/corollaries of our results.
@@ -14,6 +15,8 @@ Finally, in~\Cref{sec:momemts}, we generalize our result for expectation to othe
 
 \revision{
 \subsection{Cost Model, Query Plans, and Runtime}
+As in the introduction, we could consider polynomials to be represented as an expression tree.
+However, they do not capture many of the compressed polynomial representations that we can get from query processing algorithms on bags, including the recent work on worst-case optimal join algorithms~\cite{ngo-survey,skew}, factorized databases~\cite{factorized-db}, and FAQ~\cite{DBLP:conf/pods/KhamisNR16}. Intuitively, the main reason is that an expression tree does not allow for `sharing' of intermediate results, which is crucial for these algorithms (and other query processing methods as well).
 }
 
 \label{sec:circuits}

ra-to-poly.tex

@@ -96,15 +96,13 @@ We first formally define circuits, an encoding of polynomials that we use throug
 For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\oldstuff{\begin{Definition}[Expression Tree]\label{def:express-tree}
-Consider a vector of variables $\vct{X}$.
-  An expression tree $\etree$ over $\vct{X}$ is a binary %an ADT logically viewed as an n-ary
-tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either from the set $\mathbb{R}$ $(\tnum)$ or from the set of monomials $(\var)$.  The members of $\etree$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\etree$ (i.e. one of $\{+, \times, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of $\etree$'s children where $\etree_\lchild$ is the left child and $\etree_\rchild$ the right child.
-\end{Definition}}
+%\oldstuff{\begin{Definition}[Expression Tree]\label{def:express-tree}
+%Consider a vector of variables $\vct{X}$.
+%  An expression tree $\etree$ over $\vct{X}$ is a binary %an ADT logically viewed as an n-ary
+%tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either from the set $\mathbb{R}$ $(\tnum)$ or from the set of monomials $(\var)$.  The members of $\etree$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\etree$ (i.e. one of $\{+, \times, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of $\etree$'s children where $\etree_\lchild$ is the left child and $\etree_\rchild$ the right child.
+%\end{Definition}}
 
 \revision{
-As in the introduction, we could consider polynomials to be represented as an expression tree.
-However, they do not capture many of the compressed polynomial representations that we can get from query processing algorithms on bags, including the recent work on worst-case optimal join algorithms~\cite{ngo-survey,skew}, factorized databases~\cite{factorized-db}, and FAQ~\cite{DBLP:conf/pods/KhamisNR16}. Intuitively, the main reason is that an expression tree does not allow for `sharing' of intermediate results, which is crucial for these algorithms (and other query processing methods as well).
 
 We represent query polynomials via {\em arithmetic circuits}~\cite{arith-complexity}, a standard way to represent polynomials over fields (particularly in the field of algebraic complexity) that we use for polynomials over $\mathbb N$ in the obvious way.
 
@@ -137,7 +135,7 @@ Denote \revision{$\polyf(\circuit)$}~ to be the function from circuit \revision{
 \end{equation*}
 \end{Definition}
 
-Note that $\circuit$ need not encode an expression in standard monomial basis.  For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$.
+Note that $\circuit$ need not encode an expression in standard monomial basis, while as stated previously a polynomial is considered to be in SMB, and the output of \polyf($\cdot$) is therefore in SMB.  For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$.
 
 \oldstuff{
 \begin{Definition}[Expression Tree Set]\label{def:express-tree-set}$\etreeset{\smb}$ is the set of all possible expression trees $\etree$, such that $poly(\etree) = \poly(\vct{X})$.
@@ -150,7 +148,7 @@ $\circuitset{\smb}$ is the set of all possible circuits $\circuit$ such that $\p
 }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-For our running example, $\circuitset{\smb} = \{2X^2 + 3XY - 2Y^2, (X + 2Y)(2X - Y), X(2X - Y) + 2Y(2X - Y), 2X(X + 2Y) - Y(X + 2Y)\}$.  Note that ~\Cref{def:express-tree-set} implies that \revision{$\circuit \in \circuitset{\polyf(\circuit)}$}.
+For our running example, $\circuitset{\smb} \supset \{2X^2 + 3XY - 2Y^2, (X + 2Y)(2X - Y), X(2X - Y) + 2Y(2X - Y), 2X(X + 2Y) - Y(X + 2Y)\}$.  Note that ~\Cref{def:circuit-set} implies that \revision{$\circuit \in \circuitset{\polyf(\circuit)}$}.
 }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \medskip