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\subsection{Problem Definition}\label{sec:expression-trees}

We first formally define circuits, an encoding of polynomials that we use throughout the paper.  Since we are particularly using \emph{lineage} circuits, we drop the term lineage and only refer to them as circuits.
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For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.

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.

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\begin{Definition}[Circuit]\label{def:circuit}
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$.  The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
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Each internal node in a circuit $\circuit$ has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the type of value stored in the gate (one of $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} is the list of the gate's inputs. The source gates have an additional member \val, which holds the value stored (constant or variable).  We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input of a sink of circuit $\circuit$.
%The member \degval holds the degree of \circuit.
When the underlying DAG is a tree (with edges pointing towards the root), we will refer to the structure as an expression tree \etree.  Note that in such a case, the root of \etree is analogous to the sink of \circuit.
\end{Definition}
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As stated in \Cref{def:circuit}, every internal node has at most two in-edges, is labeled as an addition or a multiplication node, and has no limit on its outdegree.
Note that if we limit the outdegree to one, then we get expression trees.
We ignore the fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{We omit degree here too, which {\emph I think} is used only in appendix proofs.}


\begin{Example}
The circuit \circuit in \Cref{fig:circuit-express-tree} encodes the polynomial $XY + WZ$.  Note that circuit \circuit encodes a tree, with edges pointing towards the root.
\end{Example}

\begin{figure}[t]
	\begin{subfigure}[b]{0.45\linewidth}
		\centering
		\begin{tikzpicture}[thick]
			\node[tree_node] (a1) at (0, 0){$\boldsymbol{X}$};
			\node[tree_node] (b1) at (1, 0){$\boldsymbol{Y}$};
			\node[tree_node] (c1) at (2, 0){$\boldsymbol{W}$};
			\node[tree_node] (d1) at (3, 0){$\boldsymbol{Z}$};

			\node[tree_node] (a2) at (0.5, 1){$\boldsymbol{\circmult}$};
			\node[tree_node] (b2) at (2.5, 1){$\boldsymbol{\circmult}$};

			\node[tree_node] (a3) at (1.5, 2){$\boldsymbol{\circplus}$};

			\draw[->] (a1) -- (a2);
			\draw[->] (b1) -- (a2);
			\draw[->] (c1) -- (b2);
			\draw[->] (d1) -- (b2);
			\draw[->] (a2) -- (a3);
			\draw[->] (b2) -- (a3);
			\draw[->] (a3) -- (1.5, 2.5);
		\end{tikzpicture}
		\caption{Circuit encoding $XY + WZ$, a special case of an expression tree}
		\label{fig:circuit-express-tree}
	\end{subfigure}
	\hspace{5mm}
	\begin{subfigure}[b]{0.45\linewidth}
		\centering
		\begin{tikzpicture}[thick]
			\node[tree_node] (a1) at (0, 0) {$\boldsymbol{X}$};
			\node[tree_node] (b1) at (1.5, 0) {$\boldsymbol{2}$};
			\node[tree_node] (c1) at (3, 0) {$\boldsymbol{Y}$};
			\node[tree_node] (d1) at (4.5, 0) {$\boldsymbol{-1}$};

			\node[tree_node] (a2) at (0.75, 0.75) {$\boldsymbol{\circmult}$};
			\node[tree_node] (b2) at (2.25, 0.75) {$\boldsymbol{\circmult}$};
			\node[tree_node] (c2) at (3.75, 0.75) {$\boldsymbol{\circmult}$};

			\node[tree_node] (a3) at (0.55, 1.5) {$\boldsymbol{\circplus}$};
			\node[tree_node] (b3) at (3.75, 1.5) {$\boldsymbol{\circplus}$};

			\node[tree_node] (a4) at (2.25, 2.25) {$\boldsymbol{\circmult}$};

			\draw[->] (a1) -- (a2);
			\draw[->] (a1) -- (a3);
			\draw[->] (b1) -- (a2);
			\draw[->] (b1) -- (b2);
			\draw[->] (c1) -- (c2);
			\draw[->] (c1) -- (b2);
			\draw[->] (d1) -- (c2);
			\draw[->] (a2) -- (b3);
			\draw[->] (b2) -- (a3);
			\draw[->] (c2) -- (b3);
			\draw[->] (a3) -- (a4);
			\draw[->] (b3) -- (a4);
			\draw[->] (a4) -- (2.25, 2.75);
		\end{tikzpicture}
		\caption{Circuit encoding of $(X + 2Y)(2X - Y)$}
		\label{fig:circuit}
	\end{subfigure}
	\caption{Example circuit encodings}
\end{figure}


The semantics of circuits follows the obvious interpretation.  We next define its relationship with polynomials formally:
\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corresponding polynomial (in \abbrSMB).\footnote{Recall our assumption that unless otherwise mentioned, all polynomials are considered in $\abbrSMB$.}  $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
\begin{equation*}
	\polyf(\circuit) = \begin{cases}
					\polyf(\circuit_\lchild) + \polyf(\circuit_\rchild)			&\text{ if \circuit.\type } = \circplus\\
					\polyf(\circuit_\lchild) \cdot \polyf(\circuit_\rchild)		&\text{ if \circuit.\type } = \circmult\\
					\circuit.\val									&\text{ if \circuit.\type } = \var \text{ OR } \tnum.
				\end{cases}
\end{equation*}
\end{Definition}

Note that $\circuit$ need not encode an expression in SMB.  For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = 2X^2+3XY-2Y^2$.

\begin{Definition}[Circuit Set]\label{def:circuit-set}
$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.\footnote{Again, the representation of $\polyX$ is $\abbrSMB$.}
\end{Definition}

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The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$.  One can think of $\circuitset{\polyX}$ as the infinite set of circuits each of which equal $\polyX$ when represented in $\abbrSMB$.
Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\circuit)}$.

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\medskip

\noindent We are now ready to formally state our \textbf{main problem}.
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\begin{Definition}[The Expected Result Multiplicity Problem]\label{def:the-expected-multipl}
Let $\vct{X} = (X_1, \ldots, X_n)$, and $\pxdb$ be an $\semNX$-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X}  \to \{0,1\}$, $\query$ an n-ary query, and $t$ an n-ary tuple.
  The \expectProblem is defined as follows:\\[-7mm]
\begin{center}
\textbf{Input}: A circuit $\circuit \in \circuitset{\polyX}$ for $\polyX = \query(\pxdb)(t)$
\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
\end{center}
\end{Definition}
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