53 lines
3.5 KiB
TeX
53 lines
3.5 KiB
TeX
%root: main.tex
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%!TEX root = ./main.tex
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\iffalse
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\begin{Example}\label{example:qtilde}
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Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blocks. The expanded derivation for $\rpoly(X, Y)$ is
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\begin{align*}
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(&X^2 + 2XY + Y^2 \mod X^2 - X) \mod Y^2 - Y\\
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= ~&X + 2XY + Y^2 \mod Y^2 - Y\\
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= ~& X + 2XY + Y
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\end{align*}
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\end{Example}
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\fi
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let $\abs{\poly'}$ be the number of operators in $\poly'$. Then:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Corollary}\label{cor:expct-sop}
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If $\poly'$ is a \abbrOneBIDB lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $ \expct_{\vct{W} \sim \pdassign'}\pbox{\poly'\inparen{\vct{W}}}$ % = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$
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can be computed in $\bigO{\abs{\poly'}}$ time.
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\end{Corollary}
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% \subsubsection{Possible World Semantics}\label{subsub:possible-world-sem}
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% In this section, we show how the traditional possible worlds semantics corresponds to our setup. Readers can safely skip this part without missing anything vital to the results of this paper.
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% Queries over probabilistic databases are traditionally viewed as being evaluated using the so-called possible world semantics. A general bag-\abbrPDB can be defined as the pair $\pdb = \inparen{\Omega, \bpd}$ where $\Omega$ is the set of possible worlds represented by $\pdb$ and $\bpd$ the probability distribution over $\Omega$. 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: $\probOf\pbox{\omega\in\Omega} = \sum_{\omega'\in\Omega,\\\query\inparen{\omega'}=\query\inparen{\omega}}\probOf\pbox{\omega'}$.
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% Suppose that $\pdb'$ is a reduced \abbrOneBIDB from \abbrCTIDB $\pdb$ as defined by~\Cref{prop:ctidb-reduct}. Instead of looking only at the possible worlds of $\pdb'$, one can consider the set of all worlds, including those that cannot exist due to, e.g., disjointness. Since $\abs{\tupset} = \numvar$ the all worlds set can be modeled by $\worldvec\in \{0, 1\}^{\numvar\bound}$, such that $\worldvec_{\tup, j} \in \worldvec$ represents whether or not the multiplicity of $\tup$ is $j$ (\emph{here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_{\numvar'}$, where $\numvar'=\sum_{\tup\in\tupset}\abs{\block_\tup}$}).
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% \footnote{
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% In this example, $\abs{\block_\tup} = \bound$ for all $\tup$.
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% }
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% We can denote a probability distribution over all $\worldvec \in \{0, 1\}^{\numvar\bound}$ as $\bpd'$. When $\bpd'$ is the one induced from each $\prob_{\tup, j}$ while assigning $\probOf\pbox{\worldvec} = 0$ for any $\worldvec$ with $\worldvec_{\tup, j}, \worldvec_{\tup, j'} \neq 0$ 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, and thus query results.
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% \Cref{subsec:supp-mat-ti-bi-def} has more details. \medskip
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% We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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