141 lines
7.4 KiB
TeX
141 lines
7.4 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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\subsection{Problem Definition}\label{sec:expression-trees}
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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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%
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For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[Circuit]\label{def:circuit}
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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 and (the single) sink gate of $\circuit$ (corresponding to the result tuple $t$) have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
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%
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Each node in a circuit $\circuit$ has the following members: \type, \val, \vpartial, \vari{input}, \degval and \vari{Lweight}, \vari{Rweight}, where \type is the type of value stored in the gate (one of $\{\circplus, \circmult, \var, \tnum\}$, \val is the value stored (a constant or variable), and \vari{input} is the list of the gate's inputs. We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input of the sink of circuit $\circuit$.
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%The member \degval holds the degree of \circuit.
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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.
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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.
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Note that if we limit the outdegree to one, then we get expression trees.
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We ignore the fields \vari{partial}, \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.}
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\begin{Example}
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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.
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\end{Example}
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\begin{figure}[t]
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\begin{subfigure}[b]{0.45\linewidth}
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\centering
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\begin{tikzpicture}[thick]
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\node[tree_node] (a1) at (0, 0){$\boldsymbol{X}$};
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\node[tree_node] (b1) at (1, 0){$\boldsymbol{Y}$};
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\node[tree_node] (c1) at (2, 0){$\boldsymbol{W}$};
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\node[tree_node] (d1) at (3, 0){$\boldsymbol{Z}$};
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\node[tree_node] (a2) at (0.5, 1){$\boldsymbol{\circmult}$};
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\node[tree_node] (b2) at (2.5, 1){$\boldsymbol{\circmult}$};
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\node[tree_node] (a3) at (1.5, 2){$\boldsymbol{\circplus}$};
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\draw[->] (a1) -- (a2);
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\draw[->] (b1) -- (a2);
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\draw[->] (c1) -- (b2);
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\draw[->] (d1) -- (b2);
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\draw[->] (a2) -- (a3);
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\draw[->] (b2) -- (a3);
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\draw[->] (a3) -- (1.5, 2.5);
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\end{tikzpicture}
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\caption{Circuit encoding $XY + WZ$, a special case of an expression tree}
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\label{fig:circuit-express-tree}
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\end{subfigure}
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\hspace{5mm}
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\begin{subfigure}[b]{0.45\linewidth}
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\centering
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\begin{tikzpicture}[thick]
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\node[tree_node] (a1) at (0, 0) {$\boldsymbol{X}$};
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\node[tree_node] (b1) at (1.5, 0) {$\boldsymbol{2}$};
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\node[tree_node] (c1) at (3, 0) {$\boldsymbol{Y}$};
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\node[tree_node] (d1) at (4.5, 0) {$\boldsymbol{-1}$};
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\node[tree_node] (a2) at (0.75, 0.75) {$\boldsymbol{\circmult}$};
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\node[tree_node] (b2) at (2.25, 0.75) {$\boldsymbol{\circmult}$};
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\node[tree_node] (c2) at (3.75, 0.75) {$\boldsymbol{\circmult}$};
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\node[tree_node] (a3) at (0.55, 1.5) {$\boldsymbol{\circplus}$};
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\node[tree_node] (b3) at (3.75, 1.5) {$\boldsymbol{\circplus}$};
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\node[tree_node] (a4) at (2.25, 2.25) {$\boldsymbol{\circmult}$};
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\draw[->] (a1) -- (a2);
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\draw[->] (a1) -- (a3);
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\draw[->] (b1) -- (a2);
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\draw[->] (b1) -- (b2);
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\draw[->] (c1) -- (c2);
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\draw[->] (c1) -- (b2);
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\draw[->] (d1) -- (c2);
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\draw[->] (a2) -- (b3);
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\draw[->] (b2) -- (a3);
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\draw[->] (c2) -- (b3);
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\draw[->] (a3) -- (a4);
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\draw[->] (b3) -- (a4);
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\draw[->] (a4) -- (2.25, 2.75);
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\end{tikzpicture}
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\caption{Circuit encoding of $(X + 2Y)(2X - Y)$}
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\label{fig:circuit}
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\end{subfigure}
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\caption{Example circuit encodings}
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\end{figure}
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The semantics of circuits follows the obvious interpretation. We next define its relationship with polynomials formally:
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\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
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Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corresponding polynomial. $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
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\begin{equation*}
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\polyf(\circuit) = \begin{cases}
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\polyf(\circuit_\lchild) + \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circplus\\
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\polyf(\circuit_\lchild) \cdot \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circmult\\
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\circuit.\val &\text{ if \circuit.\type } = \var \text{ OR } \tnum.
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\end{cases}
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\end{equation*}
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\end{Definition}
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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 $\polyf(\circuit) = 2X^2+3XY-2Y^2$.\footnote{As stated previously, unless otherwise mentioned all polynomials are considered in the $\abbrSMB$ representation, and this implies that the output of $\polyf\inparen{\cdot}$ is indeed $\abbrSMB$.}
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\begin{Definition}[Circuit Set]\label{def:circuit-set}
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$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.\footnote{Again, the representation of $\polyX$ is $\abbrSMB$.}
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 $\abbrSMB$.
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Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\circuit)}$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\medskip
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\noindent We are now ready to formally state our \textbf{main problem}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[The Expected Result Multiplicity Problem]\label{def:the-expected-multipl}
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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.
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The \expectProblem is defined as follows:\\[-7mm]
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\begin{center}
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\textbf{Input}: A circuit $\circuit \in \circuitset{\polyX}$ for $\polyX = \query(\pxdb)(t)$
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\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
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\end{center}
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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