Finished S.2 pass.
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@ -41,34 +41,34 @@ For these algorithms, $\jointime{R_1, \ldots, R_n}$ is linear in the {\em AGM bo
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\noindent\resizebox{1\linewidth}{!}{
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\begin{minipage}{1.0\linewidth}
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\begin{align*}
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\qruntime{R,\db} & = |\db.R| &
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\qruntime{\sigma \query, \db} & = \qruntime{\query,\db} &
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\qruntime{\pi \query, \db} & = \qruntime{\query,\db} + \abs{\query(\db)}
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\qruntimenoopt{R,\db} & = |\db.R| &
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\qruntimenoopt{\sigma \query, \db} & = \qruntimenoopt{\query,\db} &
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\qruntimenoopt{\pi \query, \db} & = \qruntimenoopt{\query,\db} + \abs{\query(\db)}
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\end{align*}\\[-15mm]
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\begin{align*}
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\qruntime{\query \cup \query', \db} & = \qruntime{\query, \db} +
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\qruntime{\query', \db} +
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\qruntimenoopt{\query \cup \query', \db} & = \qruntimenoopt{\query, \db} +
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\qruntimenoopt{\query', \db} +
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\abs{\query(D)}+\abs{\query'(D)} \\
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\qruntime{\query_1 \bowtie \ldots \bowtie \query_m, \db}
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& = \qruntime{\query_1, \db} + \ldots +
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\qruntime{\query_m,\db} +
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\qruntimenoopt{\query_1 \bowtie \ldots \bowtie \query_m, \db}
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& = \qruntimenoopt{\query_1, \db} + \ldots +
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\qruntimenoopt{\query_m,\db} +
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\jointime{\query_1(\db), \ldots, \query_m(\db)}
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\end{align*}
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\end{minipage}
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}\\
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Under this model, an $\raPlus$ query $\query$ evaluated over database $\db$ has runtime $O(\qruntime{Q,\db})$.
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Under this model, an $\raPlus$ query $\query$ evaluated over database $\db$ has runtime $O(\qruntimenoopt{Q,\db})$.
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We assume that full table scans are used for every base relation access. We can model index scans by treating an index scan query $\sigma_\theta(R)$ as a base relation.
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%Observe that
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% () .\footnote{This claim can be verified by e.g. simply looking at the {\em Generic-Join} algorithm in~\cite{skew} and {\em factorize} algorithm in~\cite{factorized-db}.} It can be verified that the above cost model on the corresponding $\raPlus$ join queries correctly captures the runtime of current best known .
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Finally, \Cref{lem:circ-model-runtime} and \Cref{lem:tlc-is-the-same-as-det} show that for any $\raPlus$ query $\query$ and $\dbbase$, there exists a circuit $\circuit^*$ such that $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit^*)$ and $|\circuit^*|$ are both $O(\qruntime{Q, \dbbase})$. Recall we assumed these two bounds when we moved from \Cref{prob:big-o-joint-steps} to \Cref{prob:intro-stmt}.
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Finally, \Cref{lem:circ-model-runtime} and \Cref{lem:tlc-is-the-same-as-det} show that for any $\raPlus$ query $\query$ and $\dbbase$, there exists a circuit $\circuit^*$ such that $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit^*)$ and $|\circuit^*|$ are both $O(\qruntimenoopt{Q, \dbbase})$. Recall we assumed these two bounds when we moved from \Cref{prob:big-o-joint-steps} to \Cref{prob:intro-stmt}.
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%
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%We now make a simple observation on the above cost model:
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%\begin{proposition}
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%\label{prop:queries-need-to-output-tuples}
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%The runtime $\qruntime{Q}$ of any query $Q$ is at least $|Q|$
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%The runtime $\qruntimenoopt{Q}$ of any query $Q$ is at least $|Q|$
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%\end{proposition}
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -79,7 +79,7 @@ Finally, \Cref{lem:circ-model-runtime} and \Cref{lem:tlc-is-the-same-as-det} sho
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% Given an $\raPlus$ query $\query$ over a \ti $\pdb$ with \dbbaseName $\dbbase$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pdb)$ with probability at least $1-\delta$ in time
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%
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% \[
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% O_k\left(\frac 1{\eps^2}\cdot\qruntime{Q,\dbbase}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)
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% O_k\left(\frac 1{\eps^2}\cdot\qruntimenoopt{Q,\dbbase}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)
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% \]
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%\end{Corollary}
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%Atri: The above is no longer needed
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@ -334,6 +334,7 @@
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\newcommand{\ptime}{{\sf PTIME}\xspace}
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\newcommand{\timeOf}[1]{T_{#1}}
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\newcommand{\qruntime}[1]{T^*_{det}(#1)}
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\newcommand{\qruntimenoopt}[1]{T_{det}\inparen{#1}}
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\newcommand{\jointime}[1]{T_{join}(#1)}
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\newcommand{\kmatchtime}{T_{match}\inparen{k, G}}
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@ -4,15 +4,15 @@
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%\subsection{Reduced Polynomials and Equivalences}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\AH{\Cref{def:reduced-poly} replaces this.}
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\begin{Definition}[Reduced \bi Polynomials]\label{def:reduced-bi-poly}
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Let $\poly(\vct{X})$ be a \bi-lineage polynomial.
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The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is the same as \Cref{def:reduced-poly} with the added constraint that all monomials with variables $X_{\block, i}, X_{\block, j}, i\neq j$ from the same block $\block$ are omitted.
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\end{Definition}
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%\AH{\Cref{def:reduced-poly} replaces this.}
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%\begin{Definition}[Reduced \bi Polynomials]\label{def:reduced-bi-poly}
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% Let $\poly(\vct{X})$ be a \bi-lineage polynomial.
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% The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is the same as \Cref{def:reduced-poly} with the added constraint that all monomials with variables $X_{\block, i}, X_{\block, j}, i\neq j$ from the same block $\block$ are omitted.
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%\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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Consider a $\abbrBIDB$ polynomial $\poly\inparen{\vct{X}} = X_{1, 1}X_{1, 2} + X_{1, 2}X_{2, 1}^2$. Then by \Cref{def:reduced-bi-poly}, we have that $\rpoly\inparen{\vct{X}} = X_{1, 2}X_{2, 1}$. Next, we show why the reduced form is useful for our purposes.
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%
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%Consider a $\abbrBIDB$ polynomial $\poly\inparen{\vct{X}} = X_{1, 1}X_{1, 2} + X_{1, 2}X_{2, 1}^2$. Then by \Cref{def:reduced-bi-poly}, we have that $\rpoly\inparen{\vct{X}} = X_{1, 2}X_{2, 1}$. Next, we show why the reduced form is useful for our purposes.
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%%Removing this example to save space
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\iffalse
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\begin{Example}\label{example:qtilde}
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@ -28,14 +28,13 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\AH{Reduction \emph{I think} combined with~\Cref{lem:tidb-reduce-poly} should replace this. In fact~\Cref{lem:tidb-reduce-poly} works for also for \abbrBIDB\xplural, so, maybe still stating this, but that it follows from~\Cref{lem:tidb-reduce-poly}.}
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\begin{Lemma}\label{lem:exp-poly-rpoly}
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Let $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability distribution $\pdassign$ over $\{0,1\}^\numvar$ (the all worlds set) is induced by the probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$. As in \Cref{lem:tidb-reduce-poly} for \abbrTIDB, any \abbrBIDB-lineage polynomial $\poly(\vct{X})$ based on $\pdb$ and query $\query$ we have:
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% The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
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\begin{equation*}
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\expct_{\vct{W}\sim \pdassign}\pbox{\poly(\vct{W})} = \rpoly(\probAllTup).
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\end{equation*}
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\end{Lemma}
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%\begin{Lemma}\label{lem:exp-poly-rpoly}
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%Let $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability distribution $\pdassign$ over $\{0,1\}^\numvar$ (the all worlds set) is induced by the probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$. As in \Cref{lem:tidb-reduce-poly} for \abbrTIDB, any \abbrBIDB-lineage polynomial $\poly(\vct{X})$ based on $\pdb$ and query $\query$ we have:
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% % The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
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%\begin{equation*}
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%\expct_{\vct{W}\sim \pdassign}\pbox{\poly(\vct{W})} = \rpoly(\probAllTup).
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%\end{equation*}
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%\end{Lemma}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let $\abs{\poly}$ be the number of operators in $\poly$.
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@ -43,6 +42,21 @@ Let $\abs{\poly}$ be the number of operators in $\poly$.
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\begin{Corollary}\label{cor:expct-sop}
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If $\poly$ is a \bi-lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $\bigO{\abs{\poly}}$ time.
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\end{Corollary}
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\secrev{
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Queries over probabilistic databases are evaluated using the so-called possible world semantics. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\Omega$ is the set of query answers produced by evaluating $\query$ over each possible world $\omega\in\Omega$: $\inset{\query\inparen{\omega}: \omega\in\Omega}$.
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The result of a query is the pair $\inparen{\query\inparen{\omega}, \bpd'}$ where $\bpd'$ is a probability distribution that assigns to each possible query result the sum of the probabilites of the worlds that produce this answer: $\probOf\pbox{\omega\in\Omega} = \sum\limits_{\substack{\omega'\in\Omega,\\\query\inparen{\omega'}=\query\inparen{\omega}}}\probOf\pbox{\omega'}$.
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}
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Recall \Cref{fig:nxDBSemantics} which defines the lineage polynomial $\apolyqdt$ for any $\raPlus$ query. We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
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\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
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Given a \abbrBPDB $\pdb = (\Omega,\bpd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for arbitrary result tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
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we have (denoting $\randDB$ as the random variable over $\idb$):
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$ \expct_{\randDB \sim \bpd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
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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}.\footnote{Although \Cref{prop:expection-of-polynom} follows, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
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We focus on the problem of computing $\expct_\pdassign\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ from now on, assume implicit $\query, \tupset, \tup$, and drop them from $\apolyqdt$ (i.e., $\poly\inparen{\vct{X}}$ will denote a polynomial).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -38,37 +38,24 @@ We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\abbrCTIDB-lineage polynom
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or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \abbrCTIDB $\pdb$, and result tuple $\tup$ such that $\poly\inparen{\vct{X}} = \apolyqdt\inparen{\vct{X}}.$
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}
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\subsection{Probabilistic Databases}
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Following the typical representation of bags in production databases, for query inputs, we will use \abbrBPDB\xplural with multiplicities $\{0, 1\}$ (see \Cref{sec:gener-results-beyond} for more on this choice).
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An \textit{incomplete database} $\idb$ is a set of deterministic databases $\db$ called possible worlds.
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\secrev{
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A \abbrCTIDB $\pdb$ is a pair $\inparen{\worlds, \bpd}$ such that $\worlds$ is an incomplete database and $\bpd$ is a probability distribution over $\worlds$. Queries over probabilistic databases (and thus \abbrCTIDB\xplural) are evaluated using the so-called possible world semantics. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\worlds$ is the set of query answers produced by evaluating $\query$ over each possible world $\worldvec\in\worlds$: $\inset{\query\inparen{\worldvec}: \worldvec\in\worlds}$.
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The result of a query is the pair $\inparen{\query\inparen{\worlds}, \bpd'}$ where $\bpd'$ is a probability distribution that assigns to each possible query result the sum of the probabilites of the worlds that produce this answer: $\probOf\pbox{\worldvec\in\worlds} = \sum\limits_{\substack{\worldvec'\in\worlds,\\\query\inparen{\worldvec'}=\query\inparen{\worldvec}}}\probOf\pbox{\worldvec'}$.
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}
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\AH{Move~\Cref{prop:expection-of-polynom} to after/at the end of~\Cref{subsec:tidbs-and-bidbs}. We may have to introduce some new notation for $1$-\abbrBIDB, like the $\randDB$ in the expecatation.}
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Recall \Cref{fig:nxDBSemantics} which defines the lineage polynomial $\apolyqdt$ for any $\raPlus$ query. We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
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\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
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Given a \abbrBPDB $\pdb = (\idb,\pd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for arbitrary result tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
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we have (denoting $\randDB$ as the random variable over $\idb$):
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$ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
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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}.\footnote{Although \Cref{prop:expection-of-polynom} follows, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
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We focus on the problem of computing $\expct_\pdassign\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ from now on, assume implicit $\query, \dbbase, \tup$, and drop them from $\apolyqdt$ (i.e., $\poly\inparen{\vct{X}}$ will denote a polynomial).
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%Following the typical representation of bags in production databases, for query inputs, we will use \abbrBPDB\xplural with multiplicities $\{0, 1\}$ (see \Cref{sec:gener-results-beyond} for more on this choice).
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\subsubsection{Reduction to $1$-\abbrBIDB}
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\label{subsec:tidbs-and-bidbs}
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\secrev{
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A block independent database (\abbrBIDB) is a common probabilistic data model $\pdb=\inparen{\Omega, \pdb}$ such that the base set of tuples $\tupset = \bigcup_{\omega\in\Omega,~\tup\in\omega}\tup$ is partitioned into a set of $\numvar$ independent blocks $\inset{\inparen{\block_\tup}_{\tup\in\pbox{\numvar}}}$ such that the set of tuples $\inset{\inparen{\tup_j}_{j\in\pbox{\abs{\block}}}}$ in block $\block_\tup$ are disjoint from one another. This construction produces the set of possible worlds $\Omega$ that consists of all unique combinations of tuples in $\tupset$ with the constraint that for any $\omega\in\Omega$, no two tuples $\tup_j\neq\tup_{j'}$ from the same block $\block_\tup$ exist together. A $\bound$-\abbrBIDB has the further requirement that each block has a multiplicity of at most $c$. We present a reduction that is useful in producing our results:
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An \textit{incomplete database} $\Omega$ is a set of deterministic databases $\omega$ called possible worlds.
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\noindent\secrev{
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A \abbrCTIDB $\pdb$ is a pair $\inparen{\worlds, \bpd}$ such that $\worlds$ is an incomplete database whose set of possible worlds is the $c+1^\numvar$ tuple/multiplicity combinations for each $\tup\in\tupset$, where $\abs{\tupset} = \numvar$, $\tupset = \bigcup_{\worldvec\in\worlds,~\worldvec_{\tup}\geq 1}\tup$ is the set of possible tuples across possible worlds, and $\bpd$ is a probability distribution over $\worlds$.
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A block independent database (\abbrBIDB) is a related probabilistic data model $\pdb=\inparen{\Omega, \pdb}$ such that the base set of tuples $\tupset = \bigcup_{\omega\in\Omega,~\tup\in\omega}\tup$ is partitioned into a set of $\numvar$ independent blocks $\inset{\inparen{\block_\tup}_{\tup\in\pbox{\numvar}}}$ such that the set of tuples $\inset{\inparen{\tup_j}_{j\in\pbox{\abs{\block}}}}$ in block $\block_\tup$ are disjoint from one another. This construction produces the set of possible worlds $\Omega$ that consists of all unique combinations of tuples in $\tupset$ with the constraint that for any $\omega\in\Omega$, no two tuples $\tup_j, \tup_{j'}, j\neq j'$ from the same block $\block_\tup$ exist together. A $\bound$-\abbrBIDB has the further requirement that each block has a multiplicity of at most $c$. We present a reduction that is useful in producing our results:
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\begin{Definition}[\abbrCTIDB reduction]\label{def:ctidb-reduct}
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Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\Omega, \bpd'}$ be the \abbrOneBIDB obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup$ with $\bound$ disjoint copies, such that $\tup_j$ is annotated with variable $X_{\tup, j}$ for $j\in\pbox{\bound}$. The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement.
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Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\Omega, \bpd'}$ be the \abbrOneBIDB obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, X_{\tup, j}}_{j\in\pbox{\bound}}}$ %with $\bound$ disjoint copies, such that $\tup_j$ is annotated with variable $X_{\tup, j}$ for $j\in\pbox{\bound}$.
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The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement.
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\end{Definition}
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The base case of~\Cref{fig:nxDBSemantics} for $\pdb'$ is now $\poly\pbox{\rel,\tupset, \tup} = \sum_{j\in\pbox{\bound}}X_{\tup, j}$. Then given the disjoint requirement and the semantics for constructing the lineage polynomial over a \abbrOneBIDB, $\poly\pbox{\rel,\tupset',\tup}$ is of the same form as the reformulated polynomial $\refpoly$ of step i) from~\Cref{def:reduced-poly}, which then implies that~\Cref{lem:tidb-reduce-poly} immediately follows for \abbrOneBIDB polynomials: $\expct_{\rvworld\sim\bpd'}\pbox{\poly\inparen{\rvworld}} = \rpoly\inparen{\vct{\prob}}$.
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As previously noted, unlike $X_{\tup}\in\inset{0,\ldots,\bound}$ for $X_{\tup}\in\vars{\pdb}$, $X_{\tup, j}\in\inset{0,1}$ for $X_{\tup, j}\in\vars{\pdb'}$. In the reduced \abbrOneBIDB setting, the base case of~\Cref{fig:nxDBSemantics} now becomes $\poly\pbox{\rel,\tupset, \tup} = \sum_{j\in\pbox{\bound}}X_{\tup, j}$. Then given the disjoint requirement and the semantics for constructing the lineage polynomial over a \abbrOneBIDB, $\poly\pbox{\rel,\tupset',\tup}$ is of the same form as the reformulated polynomial $\refpoly$ of step i) from~\Cref{def:reduced-poly}, which then implies that $\rpoly$ is the reduced polynomial that results from step ii) of~\Cref{def:reduced-poly}, and further that~\Cref{lem:tidb-reduce-poly} immediately follows for \abbrOneBIDB polynomials: $\expct_{\rvworld\sim\bpd'}\pbox{\poly\inparen{\rvworld}} = \rpoly\inparen{\vct{\prob}}$.
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}
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%In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
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%%
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@ -81,13 +68,10 @@ The base case of~\Cref{fig:nxDBSemantics} for $\pdb'$ is now $\poly\pbox{\rel,\t
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Instead of looking only at the possible worlds of $\pdb$, one can consider all worlds, including those that cannot exist due to disjointness. Then all worlds set can be modeled by $\vct{\randWorld}\in \{0, 1\}^\numvar$
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\AH{We can use the new notation $\vct{W}\in\inset{0, 1}^\tupset$ here.}
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,\footnote{Here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$.} such that $\randWorld_k \in \vct{\randWorld}$ represents the presence of $\tup_{i, j}$ (where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$). We denote a probability distribution over all $\vct{\randWorld} \in \{0, 1\}^\numvar$ as $\pdassign$. When $\pdassign$ is the one induced from each $\prob_{i, j}$ while assigning $\probOf\pbox{\vct{\randWorld}} = 0$ for any $\vct{\randWorld}$ with $\randWorld_{i, j} = \randWorld_{i, k} = 1$ for any block $i$ and $j\neq k$, we end up with a bijective mapping from $\pd$ to $\pdassign$, such that each mapping is equivalent, implying the distributions are equivalent.
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Instead of looking only at the possible worlds of $\pdb;$, one can consider all worlds, including those that cannot exist due to disjointness. The all worlds set can be modeled by $\worldvec\in \{0, 1\}^{\bound\numvar}$,\footnote{Here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$.} such that $\worldvec_{\tup, j} \in \worldvec$ represents whether or not the multiplicity of $\tup$ is $j$.%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
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We denote a probability distribution over all $\worldvec \in \{0, 1\}^\numvar$ as $\bpd''$. When $\bpd''$ is the one induced from each $\prob_{\tup, j}$ while assigning $\probOf\pbox{\worldvec} = 0$ for any $\worldvec$ with $\worldvec_{\tup, j} = \worldvec_{\tup, j'} = 1$ for $j\neq j'$, we end up with a bijective mapping from $\bpd'$ to $\bpd''$, such that each mapping is equivalent, implying the distributions are equivalent.
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\Cref{subsec:supp-mat-ti-bi-def} has more details.
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\AH{Above, we need to use new notation of $\bpd$ instead of $\pd$, and we can use $\bpd'$ for the mapping discussion and note that $\bpd\equiv\bpd'$.}% explains \abbrTIDB\xplural and \abbrBIDB\xplural in greater detail.
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%%% Local Variables:
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%%% mode: latex
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Reference in New Issue