170 lines
14 KiB
TeX
170 lines
14 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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%\onecolumn
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\section{Background and Notation}\label{sec:background}
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\subsection{Probabilistic Databases (PDBs)}
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An \textit{incomplete database} $\idb$ is a set of deterministic databases $\db$ called possible worlds.
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Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\pdb$ is a pair $(\idb, \pd)$ where $\idb$ is an incomplete database and $\pd$ is a probability distribution over $\idb$. Queries over probabilistic databases are evaluated using the so-called possible world semantics. Under possible world semantics, the result of a query $\query$ over an incomplete database $\idb$ is the set of query answers produced by evaluating $\query$ over each possible world:
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\[\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}\]
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For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$ that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
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\[\forall \db \in \query(\idb): \probOf'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \probOf(\db') \]
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Note that in this work we consider multisets, i.e., each possible world is a set of multiset relations and queries are evaluated using bag semantics. We will use $\domK$-relations to model multisets. A \emph{$\domK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$. A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
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Let $\udom$ be a countable domain of values.
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Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$.
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A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.
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We review positive relational algebra semantics for $\semK$-relations below.
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Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result $t$, we treat $\query(\db)(t)$ as a random $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \probDist}[\query(\db)(t)]$:
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%
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\begin{align}\label{eq:bag-expectation}
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\expct_{\idb \sim \probDist}[\query(\db)(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \probOf(\db)
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\end{align}
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%
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Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of $t$ we expect to find in result of query $\query$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Representation System and Semantics}\label{sec:semnx-as-repr}
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\subsubsection{$\semK$-relational Query Semantics}
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For completeness, we briefly review the semantics for $\raPlus$ queries over $\semK$-relations~\cite{DBLP:conf/pods/GreenKT07}.
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We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$. In the definition shown below, we assume that tuples are of appropriate arity and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$. Furthermore, $\theta(\tup)$ denotes the (Boolean) result of evaluating condition $\theta$ over $\tup$.
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%
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\begin{align*}
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& \evald{\project_A(\rel)}{\db}(\tup) & & = & & \sum_{\tup': \project_A(\tup') = \tup} \evald{\rel}{\db}(\tup') \\
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& \evald{(\rel_1 \union \rel_2)}{\db}(\tup) & & = & & \evald{\rel_1}{\db}(\tup) \addK \evald{\rel_2}{\db}(\tup) \\
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& \evald{(\rel_1 \join \rel_2)}{\db}(\tup) & & = & & \evald{\rel_1}{\db}(\project_{\sch(\rel_1)}(\tup)) \multK \evald{\rel_2}{\db}(\project_{\sch(\rel_2)}(\tup)) \\
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& \evald{\select_\theta(\rel)}{\db}(\tup) & & = & & \begin{cases}
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\evald{\rel}{\db}(\tup) & \text{if }\theta(\tup) \\
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\zeroK & \text{otherwise}.
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\end{cases} \\
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& \evald{R}{\db}(\tup) & & = & & \rel(\tup)
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\end{align*}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{$\semNX$ as a Representation System}\label{sec:semnx-as-repr}
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Let $\semNX$ denote the set of polynomials over variables $\vct{X}$ with natural number coefficients and exponents.
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Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$, with the standard addition and multiplication of polynomials.
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We will use $\semNX$-PDB $\pxdb$, defined as the tuple $(\db, \pd)$, where $\semNX$-database $\db$ is paired with probability distribution $\pd$.
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We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-PDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{|\vct X|} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
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$\semNX$-PDBs and a function $\rmod$ from an $\semNX$-PDB to an equivalent $\semN$-PDB are both formalized in \Cref{subsec:supp-mat-background}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
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Given an $\semN$-PDB $\pdb = (\idb,\pd)$ and $\semNX$-PDB $\pxdb = (\db',\pd')$ where $\rmod(\pxdb) = \pdb$:
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\[ \expct_{\idb \sim \pd}[\query(\db)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\polyForTuple(\vct{W})} \]
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\end{Proposition}
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\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.
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This proposition shows that computing expected tuple multiplicities is equivalent to computing the expectation of a polynomial (for that tuple) from a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$.
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We focus on this problem from now on, assume an implicit result tuple, and so drop the subscript from $\polyForTuple$ (i.e., $\poly$ is used as a polynomial from now on).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{\tis and \bis}
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\label{subsec:tidbs-and-bidbs}
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In this paper, we focus on two popular forms of PDB: Block-Independent (\bi) and Tuple-Independent (\ti) PDBs.
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%
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A \bi $\pxdb = (\db, \pd)$ is an $\semNX$-PDB such that (i) every tuple is annotated with either $0$ or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pxdb$ for which $\pxdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same blocks are disjoint events.
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%
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A \emph{\ti} is a \bi where each block contains exactly one tuple.
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\Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\input{poly-form.tex}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\oldstuff{\begin{Definition}[Expression Tree]\label{def:express-tree}
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Consider a vector of variables $\vct{X}$.
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An expression tree $\etree$ over $\vct{X}$ is a binary %an ADT logically viewed as an n-ary
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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.
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\end{Definition}}
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\revision{
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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 nodes (in degree of $0$) consist of elements in either $\reals$ or $\vct{X}$. The internal and sink nodes of $\circuit$ have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
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Circuit $\circuit$ additionally has the following members: \type, \val, \vari{partial}, \vari{input}, and \vari{weight}, where \type is the type of value stored in the node $\circuit$ (i.e. one of $\{\circplus, \circmult, \var, \tnum\}$, \val is the value stored, and \vari{input} is the list of \circuit 's inputs where $\circuit_\linput$ is the left input and $\circuit_\rinput$ the right input. When the underlying DAG is a tree, we will refer to the structure as an expression tree.
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\end{Definition}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We ignore the remaining fields (\vari{partial} and \vari{weight}) until \Cref{sec:algo}. Also note that the out degree of any internal node can grow with the circuit size.
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The semantics of \revision{circuits} ~follows the obvious interpretation. We \revision{next} define \revision{the realtionship with polynomials } formally:
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\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
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Denote \revision{$\polyf(\circuit)$}~ to be the function from circuit \revision{$\circuit$}~ to its corresponding polynomial. $\polyf(\cdot)$ is recursively defined on \revision{$\circuit$}~ as follows, with addition and multiplication following the standard interpretation for polynomials:
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\begin{equation*}
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\polyf(\revision{\circuit}) = \begin{cases}
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\polyf(\revision{\circuit_\lchild}) + \polyf(\revision{\circuit_\rchild}) &\text{ if \revision{\circuit}.\type } = \revision{\circplus}\\
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\polyf(\revision{\circuit_\lchild}) \cdot \polyf(\revision{\circuit_\rchild}) &\text{ if \revision{\circuit}.\type } = \revision{\circmult}\\
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\revision{\circuit.\val} &\text{ if \revision{\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 standard monomial basis. For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$.
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\oldstuff{
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\begin{Definition}[Expression Tree Set]\label{def:express-tree-set}$\etreeset{\smb}$ is the set of all possible expression trees $\etree$, such that $poly(\etree) = \poly(\vct{X})$.
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\end{Definition}
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\revision{
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\begin{Definition}[Circuit Set]\label{def:circuit-set}
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$\circuitset{\smb}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \poly(\vct{X})$.
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\end{Definition}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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{$\circuit \in \circuitset{\polyf(\circuit)}$}.
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}
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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 $\pdb$ 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:
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% \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.}
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% \BG{Correct}
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\\\hspace*{5mm}\textbf{Input}: A \revision{circuit $\circuit \in \circuitset{\smb}$}~ for $\poly(\vct{X}) = \query(\pxdb)(t)$
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\\\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% When $\idb$ is a probabilistic database, $\idb$ can be viewed as a two-tuple $(\wSet, \pd)$, where $\wSet$ as noted, is the set of possible worlds, and $\pd$ is a probability distribution over $\wSet$.
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% The possible worlds semantics gives a framework for how to think about running queries over $\idb$. Given a query $\query$, $\query$ is deterministically run over each $\db \in \idb$, and the output of $\query(\idb)$ is defined as the set of results (worlds) from running $\query$ over each $\db_i \in \idb$. We write this formally as,
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% \[\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}.\]
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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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