Changes to S 2.4 (from etrees to circuits)

macros.tex

@@ -267,8 +267,24 @@
 %\newcommand{\SF}[1]{}
 %\newcommand{\BG}[1]{}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%God Loves You Aaron Huber! <3
+%Revision Parameters
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\revision}[1]{\color{blue}{#1}\color{black}}
+\newcommand{\oldstuff}[1]{\color{gray}{#1}\color{black}}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%God Loves You Aaron Huber! <3
+%Circuit Notation
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\circuit}{\vari{C}}
+\newcommand{\circuitset}[1]{\vari{CSet}(#1)}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%Sets
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\reals}{\mathbb{R}}
 %
 %borrowed from Su and Boris; Ihave them here for reference purposes.
 %needs to be cleaned up

main.tex

@@ -49,6 +49,7 @@ sensitive=true
 \usepackage{hyperref}
 \usepackage{url}
 \usepackage{cleveref}
+\usepackage{color}
 % \usepackage{bbold}
 \graphicspath{ {figures/} }
 

ra-to-poly.tex

@@ -95,16 +95,24 @@ We first formally define expression trees, an encoding of polynomials that we us
 For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{Definition}[Expression Tree]\label{def:express-tree}
+\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{
+\begin{Definition}[Circuit]\label{def:circuit}
+A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source nodes consist of elements in either $\reals$ or $\vct{X}$.  The internal nodes and the sink node of $\circuit$ have binary input and are either AND ($\otimes$) or OR ($\oplus$) - gates.  Further, the outdegree of any internal node can grow with the circuit size.
+
+The members of $\circuit$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\circuit$ (i.e. one of $\{OR-gate, AND-gate, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of \circuit 's children where $\circuit_\lchild$ is the left child and $\circuit_\rchild$ the right child.
 \end{Definition}
+}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 We ignore the remaining fields (\vari{partial} and \vari{weight}) until \Cref{sec:algo}. 
-The semantics of expression trees follow the obvious interpretation; We define them formally with \Cref{def:poly-func} in \Cref{sec:algo}.
-Note that $\etree$ need not encode an expression in standard monomial basis.  For instance, $\etree$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$.
+The semantics of \revision{circuits} ~follows the obvious interpretation; We define \revision{the realtionship with polynomials } formally with \Cref{def:poly-func} in \Cref{sec:algo}.
+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 $(X + 2Y)(2X - Y)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\begin{Definition}[poly$(\cdot)$]\label{def:poly-func}
@@ -130,12 +138,19 @@ Note that $\etree$ need not encode an expression in standard monomial basis.  Fo
 %Specifically, when adding two monomials whose variables and respective exponents agree, the coefficients corresponding to the monomials are added and their sum is multiplied to the monomial.  Multiplication here is denoted by concatenation of the monomial and coefficient.  When two monomials are multiplied, the product of each corresponding coefficient is computed, and the variables in each monomial are multiplied, i.e., the exponents of like variables are added.  Again we notate this by the direct product of coefficient product and all disitinct variables in the two monomials, with newly computed exponents.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\oldstuff{
 \begin{Definition}[Expression Tree Set]\label{def:express-tree-set}$\etreeset{\smb}$ is the set of all possible expression trees $\etree$, such that $poly(\etree) = \poly(\vct{X})$.
 \end{Definition}
+
+\revision{
+\begin{Definition}[Circuit Set]\label{def:circuit-set}
+$\circuitset{\smb}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \poly(\vct{X})$.
+\end{Definition}
+}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-For our running example, $\etreeset{\smb} = \{2X^2 + 3XY - 2Y^2, (X + 2Y)(2X - Y), X(2X - Y) + 2Y(2X - Y), 2X(X + 2Y) - Y(X + 2Y)\}$.  Note that \cref{def:express-tree-set} implies that $\etree \in \etreeset{poly(\etree)}$.
-
+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{$\circuit \in \circuitset{\polyf(\circuit)}$}.
+}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \medskip
 
@@ -146,7 +161,7 @@ Let $\vct{X} = (X_1, \ldots, X_n)$, and $\pdb$ be an $\semNX$-PDB over $\vct{X}$
   The \expectProblem is defined as follows:
   % \AH{I think we mean $\poly(\vct{X}) = \query(\pxdb)(t)$ instead of $\poly(\vct{X}) = \query(\pdb)(t)$.  I changed the following to reflect this.}
   % \BG{Correct}
-\\\hspace*{5mm}\textbf{Input}: An expression tree $\etree \in \etreeset{\smb}$ for $\poly(\vct{X}) = \query(\pxdb)(t)$
+\\\hspace*{5mm}\textbf{Input}: A \revision{circuit $\circuit \in \circuitset{\smb}$}~ for $\poly(\vct{X}) = \query(\pxdb)(t)$
 \\\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%