paper-BagRelationalPDBsAreHard/pwsem.tex

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\iffalse
\begin{Example}\label{example:qtilde}
Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blocks.  The expanded derivation for $\rpoly(X, Y)$ is
\begin{align*}
(&X^2 + 2XY + Y^2 \mod X^2 - X) \mod Y^2 - Y\\
= ~&X + 2XY + Y^2 \mod Y^2 - Y\\
= ~& X + 2XY + Y
\end{align*}
\end{Example}
\fi
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Let $\abs{\poly'}$ be the number of operators in $\poly'$. Then:

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\begin{Corollary}\label{cor:expct-sop}
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)$ 
can be computed in $\bigO{\abs{\poly'}}$ time.
\end{Corollary}

% \subsubsection{Possible World Semantics}\label{subsub:possible-world-sem}
% 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.

% 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 $\omega\in\Omega$: $\inset{\query\inparen{\omega}: \omega\in\Omega}$.
% 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: $\probOf\pbox{\omega\in\Omega} = \sum_{\omega'\in\Omega,\\\query\inparen{\omega'}=\query\inparen{\omega}}\probOf\pbox{\omega'}$.


% 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 $\numvar'=\sum_{\tup\in\tupset}\abs{\block_\tup}$}).
% \footnote{
% In this example, $\abs{\block_\tup} = \bound$ for all $\tup$.
% }
%   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.
% \Cref{subsec:supp-mat-ti-bi-def} has more details. \medskip


% We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.





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