80 lines
5.9 KiB
TeX
80 lines
5.9 KiB
TeX
%root: main.tex
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%!TEX root = ./main.tex
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%\onecolumn
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%\subsection{Reduced Polynomials and Equivalences}
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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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%%Removing this example to save space
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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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%\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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Let $\abs{\poly}$ be the number of operators in $\poly$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Corollary}\label{cor:expct-sop}
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If $\poly$ is a $1$-\abbrBIDB 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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\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{def: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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}%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
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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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\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 $\Omega$):
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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