Restructuring S.2.
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poly-form.tex
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poly-form.tex
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@ -6,11 +6,11 @@
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We now introduce some terminology % for polynomials
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and develop a reduced form (a closed form of the polynomial's expectation) for polynomials over probability distributions derived from a \bi or \ti.
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%We will use $(X + Y)^2$ as a running example.
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Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ is formally defined as (with $c_\vct{i} \in \domN$):
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\AH{My attempt to clear up any confusion in the ambiguity of $c_{\vct{i}}$. We may want to say that $\domain\inparen{c_\vct{i}} = \domR$ instead?}
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Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ with highest power $B <\infty$\footnote{The standard definition of polynomials requires a finite number of terms.} and $c_\vct{i} \in \domN$ is formally defined as: %(with $c_\vct{i} \in \domN$):
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\AH{Do we want to say that $\domain\inparen{c_\vct{i}} = \domR$ instead? I've only seen the $\semNX$ use case. Is there a legitimate use case for real valued coefficients?}
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\begin{equation}
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\label{eq:sop-form}
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\poly\inparen{X_1,\dots,X_n}=\sum_{\vct{d}=(d_1,\dots,d_n)\in \semN^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i}.
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\poly\inparen{X_1,\dots,X_n}=\sum_{\vct{d}\in\{0,\ldots,B\}^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i}.
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -26,60 +26,55 @@ When it is unclear, we use $\smbOf{\poly}$ to denote the \abbrSMB form of a poly
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The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such that $c_{(d_1,\dots,d_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\AH{The example needs to change to avoid ambiguity between different definitions of \emph{degree}. Our definition is the one we want, since we two different tuples join without any tuples joining on themselves, and the degree would be $2$, rather than $1$, for example.}
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The degree of the polynomial $X^2+2XY+Y^2$ is $2$.
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Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins in any clause of the UCQ query that created it.
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In this paper we consider only finite degree polynomials.
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We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a \AH{Which formalism? UCQ?}$\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\poly\inparen{\vct{X}} = \query(\pxdb)(\tup)$.
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As an example, the degree of the polynomial $X^2+2XY^2+Y^2$ is $3$, since the largest sum of any monomial's exponents is that of the monomial $XY^2$.
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Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins in any clause of the $\raPlus$ query that created it.
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We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \bi (\ti) $\pdb$, and tuple $\tup$ such that $\poly\inparen{\vct{X}} = \query(\pdb)(\tup)$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\AH{There was reviewer disgruntlement with how we discussed modding and all in the next few definitions. Need to revisit these comments and adjust accordingly.}
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\begin{Definition}[Modding with a set]\label{def:mod-set}
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Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
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\end{Definition}
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For example for a set of polynomials $S=\inset{X^2-X, Y^2-Y}$, taking the polynomial $2X^2 + 3XY - 2Y^2\mod S$ yields $2X+3XY-2Y$.
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%
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\begin{Definition}[$\mathcal B$, $\mathcal T$]\label{def:mod-set-polys}
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Given the set of BIDB variables $\inset{X_{i,j}}$, define
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\setlength\parindent{0pt}
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\vspace*{-3mm}
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{\small
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\begin{tabular}{@{}l l}
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\begin{minipage}[b]{0.45\linewidth}
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\centering
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\begin{equation*}
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\mathcal{B}=\comprehension{X_{i,j}\cdot X_{i,j'}}{i \in [\ell], j\neq j' \in [~\abs{\block_i}~]}
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\end{equation*}
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\end{minipage}%
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\hspace{13mm}
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&
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\begin{minipage}[b]{0.45\linewidth}
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\centering
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\begin{equation*}
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\mathcal{T}=\comprehension{X_{i,j}^2-X_{i,j}}{i \in [\ell], j \in [~\abs{\block_i}~]}
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\end{equation*}
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\end{minipage}
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\\
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\end{tabular}
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}
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\end{Definition}
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%\begin{Definition}[Modding with a set]\label{def:mod-set}
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%Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
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%\end{Definition}
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%For example for a set of polynomials $S=\inset{X^2-X, Y^2-Y}$, taking the polynomial $2X^2 + 3XY - 2Y^2\mod S$ yields $2X+3XY-2Y$.
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%%
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%\begin{Definition}[$\mathcal B$, $\mathcal T$]\label{def:mod-set-polys}
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%Given the set of BIDB variables $\inset{X_{i,j}}$, define
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%
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%\setlength\parindent{0pt}
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%\vspace*{-3mm}
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%{\small
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%\begin{tabular}{@{}l l}
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% \begin{minipage}[b]{0.45\linewidth}
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% \centering
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% \begin{equation*}
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% \mathcal{B}=\comprehension{X_{i,j}\cdot X_{i,j'}}{i \in [\ell], j\neq j' \in [~\abs{\block_i}~]}
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% \end{equation*}
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% \end{minipage}%
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% \hspace{13mm}
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% &
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% \begin{minipage}[b]{0.45\linewidth}
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% \centering
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% \begin{equation*}
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% \mathcal{T}=\comprehension{X_{i,j}^2-X_{i,j}}{i \in [\ell], j \in [~\abs{\block_i}~]}
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% \end{equation*}
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% \end{minipage}
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% \\
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%\end{tabular}
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%}
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%\end{Definition}
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%%
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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: $\rpoly(\vct{X}) = \poly(\vct{X}) \mod \inparen{\mathcal{T} \cup \mathcal{B}}$
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% \begin{equation*}
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% \rpoly(\vct{X}) = \poly(\vct{X}) \mod \inparen{\mathcal{T} \cup \mathcal{B}}%X_i^2 - X_i \mod X_{\block_s, t}X_{\block_s, u}
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% \end{equation*}
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%for all $i$ in $[\numvar]$ and for all $s$ in $\ell$, such that for all $t, u$ in $[\abs{\block_s}]$, $t \neq u$.
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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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%: $\rpoly(\vct{X}) = \poly(\vct{X}) \mod \inparen{\mathcal{T} \cup \mathcal{B}}$
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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All exponents $e > 1$ in $\smbOf{\poly(\vct{X})}$ are reduced to $e = 1$ via mod $\mathcal{T}$. Performing the modulus of $\rpoly(\vct{X})$ with $\mathcal{B}$ ensures the disjoint condition of \bi, removing monomials with lineage variables from the same block.
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Consider a $\abbrBIDB$ polynomial $\poly\inparen{\vct{X}} = X_{1, 1}X_{1, 2} + X_{1, 2}X_{2, 1}$. Then by \cref{def:reduced-bi-poly}, we have that $\rpoly\inparen{\vct{X}} = X_{1, 2}X_{2, 1}$.
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%, (recall the constraint on tuples from the same block being disjoint in a \bi).% any monomial containing more than one tuple from a block has $0$ probability and can be ignored).
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%
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For the special case of \tis, the second step is not necessary since every block contains a single tuple.
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%For the special case of \tis, the second step is not necessary since every block contains a single tuple.
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%Alternatively, one can think of $\rpoly$ as the \abbrSMB of $\poly(\vct{X})$ when the product operator is idempotent.
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%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -111,14 +106,22 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
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%
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%\noindent The usefulness of this will reduction become clear in \Cref{lem:exp-poly-rpoly}.
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[Valid Worlds]
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For probability distribution $\pd$, % and its corresponding probability mass function $\probOf$,
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the set of valid worlds $\valworlds$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
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\[
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\valworlds = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
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\]
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\end{Definition}
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%NEEDS to be moved to the appendix
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{Definition}[Valid Worlds]
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%For probability distribution $\pd$, % and its corresponding probability mass function $\probOf$,
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%the set of valid worlds $\valworlds$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
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%\[
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%\valworlds = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
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%\]
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%\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%END move to appendix
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%We state additional equivalences between $\poly(\vct{X})$ and $\rpoly(\vct{X})$ in \Cref{app:subsec-pre-poly-rpoly} and \Cref{app:subsec-prop-q-qtilde}.
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\noindent Next, we show why the reduced form is useful for our purposes:
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@ -135,7 +138,7 @@ the set of valid worlds $\valworlds$ consists of all the worlds with probability
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Lemma}\label{lem:exp-poly-rpoly}
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Let $\pxdb$ be a \bi over variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ and with probability distribution $\pd$ induced by the tuple probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$ over all $\vct{w}$ in $\valworlds$. For any \bi-lineage polynomial $\poly(\vct{X})$ based on $\pxdb$ and query $\query$ we have:
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Let $\pdb$ be a \abbrBIDB over variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ and with probability distribution $\pd$ induced by the tuple probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$ over all $\vct{w}$ in $\{0, 1\}^\numvar$. For 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 \pd}\pbox{\poly(\vct{W})} = \rpoly(\probAllTup).
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@ -159,8 +162,6 @@ to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that w
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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 $\bigO{\size\inparen{\poly}}$, where $\size\inparen{\poly}$ (\Cref{def:size}) is proportional to the total number of multiplication/addition operators in $\poly$.
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\end{Corollary}
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\AH{Maybe state in terms of $\abs{\circuit}$.}
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%\AH{What if $\poly$ is not in \abbrSMB form?}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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29
prob-def.tex
29
prob-def.tex
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@ -3,26 +3,25 @@
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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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%We first formally define circuits, an encoding of polynomials that we use throughout the paper.
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%
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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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%For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
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We represent query polynomials via {\em arithmetic circuits}~\cite{arith-complexity}, a standard way to represent polynomials over fields (particularly in the field of algebraic complexity) that we use for polynomials over $\mathbb N$ in the obvious way.
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We represent query polynomials via {\em arithmetic circuits}~\cite{arith-complexity}, a standard way to represent polynomials over fields (particularly in the field of algebraic complexity) that we use for polynomials over $\mathbb N$ in the obvious way. Since we are particularly using \emph{lineage} circuits, we drop the term lineage and only refer to them as circuits.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$. The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
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%
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Each internal node in a circuit $\circuit$ has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the type of value stored in the gate (one of $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} is the list of the gate's inputs. The source gates have an additional member \val, which holds the value stored (constant or variable). We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input of a sink of circuit $\circuit$.
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%The member \degval holds the degree of \circuit.
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When the underlying DAG is a tree (with edges pointing towards the root), we will refer to the structure as an expression tree \etree. Note that in such a case, the root of \etree is analogous to the sink of \circuit.
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Each gate has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an additional member \val storing the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input.
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\end{Definition}
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When the underlying DAG is a tree (with edges pointing towards the root), the structure is an expression tree \etree. In such a case, the root of \etree is analogous to the sink of \circuit. We ignore the fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{\emph{I think} \degval is used only in appendix proofs.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As stated in \Cref{def:circuit}, every internal node has at most two in-edges, is labeled as an addition or a multiplication node, and has no limit on its outdegree.
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Note that if we limit the outdegree to one, then we get expression trees.
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We ignore the fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{We omit degree here too, which {\emph I think} is used only in appendix proofs.}
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%As stated in \Cref{def:circuit}, every internal node has at most two incoming edges, is labeled as an addition or a multiplication node, and has no limit on its outdegree.
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%Note that if we limit the outdegree to one, then we get expression trees.
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\begin{Example}
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@ -92,8 +91,7 @@ The circuit \circuit in \Cref{fig:circuit-express-tree} encodes the polynomial $
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\caption{Example circuit encodings}
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\end{figure}
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The semantics of circuits follows the obvious interpretation. We next define its relationship with polynomials formally:
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We next formally define the relationship of circuits with polynomials.
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\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
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Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corresponding polynomial (in \abbrSMB).\footnote{Recall our assumption that unless otherwise mentioned, all polynomials are considered in $\abbrSMB$.} $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
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\begin{equation*}
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@ -105,16 +103,15 @@ Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corr
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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 SMB. For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = 2X^2+3XY-2Y^2$.
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$\circuit$ need not encode $\poly\inparen{\vct{X}}$ in the same, default \abbrSMB representation. For instance, $\circuit$ could encode the factorized representation $(X + 2Y)(2X - Y)$ of $\poly\inparen{\vct{X}} = 2X^2+3XY-2Y^2$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = \poly\inparen{\vct{X}}$, the equivalent \abbrSMB representation.
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\begin{Definition}[Circuit Set]\label{def:circuit-set}
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$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.\footnote{Again, the representation of $\polyX$ is $\abbrSMB$.}
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$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. One can think of $\circuitset{\polyX}$ as the infinite set of circuits each of which equal $\polyX$ when represented in $\abbrSMB$.
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Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\circuit)}$.
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The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. One can think of $\circuitset{\polyX}$ as the infinite set of circuits where for each element \circuit, $\polyf\inparen{\circuit} = \polyX$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\medskip
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@ -122,7 +119,7 @@ Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\
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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 $\pxdb$ be an arbitrary $\semNX$-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X} \to \{0,1\}$. Fix a query $\query$ and an output tuple $\tup$.
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Let $\vct{X} = (X_1, \ldots, X_n)$ and $\pdb$ be an arbitrary \abbrBIDB-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X} \to \{0,1\}^\numvar$. Fix a query $\query$ and an output tuple $\tup$.
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The \expectProblem is defined as follows:\\[-7mm]
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\begin{center}
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\textbf{Input}: A circuit $\circuit \in \circuitset{\polyX}$ for $\polyX = \query(\pxdb)(t)$
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@ -5,7 +5,7 @@
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\subsection{Probabilistic Databases}
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The setting used in this section is primarily that of a bag-\abbrPDB query with set-\abbrPDB inputs. Recall, as noted in \cref{sec:intro-rewrite-070921}, this is not limiting.
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While the setting used in this section is primarily that of a bag-\abbrPDB query with set-\abbrPDB inputs, recall, as noted in \cref{sec:intro-rewrite-070921}, this is not limiting. All proofs are located in the appendix.
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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 the 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: $\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}$.
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@ -14,43 +14,52 @@ For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is th
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%
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\[\forall \db \in \query(\idb): \pd'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \pd(\db') \]
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Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
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We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}.
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$\semNX$-databases are functions from tuples to elements of $\semNX$, typically called annotations.
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Given an $\semNX$-database $\db$, it is common to use $\db(\tup)$ to denote the polynomial annotating tuple $\tup$ in $\db$.
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%Note that based on this definition of $\rel$, $\rel(\tup)$ is the lineage polynomial for $\tup$.
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Let $\numvar$ be the number of tuples in $\pdb$. Then, each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
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The multiplicity of $\tup \in \db$, denoted $\db(\tup)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $\tup$ on $\vct{\wElem}$.
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$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%NEEDS to be moved to the appendix.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
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%We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}.
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% $\semNX$-databases are functions from tuples to elements of $\semNX$, typically called annotations.
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%Given an $\semNX$-database $\db$, it is common to use $\db(\tup)$ to denote the polynomial annotating tuple $\tup$ in $\db$.
|
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%%Note that based on this definition of $\rel$, $\rel(\tup)$ is the lineage polynomial for $\tup$.
|
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%Let $\numvar$ be the number of tuples in $\pdb$. Then, each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
|
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%The multiplicity of $\tup \in \db$, denoted $\db(\tup)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $\tup$ on $\vct{\wElem}$.
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%$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%We will use $\semNX$-\abbrPDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$ over the assignments to $\vct{X}$.
|
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%We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-\abbrPDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{\numvar} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
|
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%$\semNX$-\abbrPDB\xplural and a function $\rmod$ (which transforms an $\semNX$-\abbrPDB to a classical bag-\abbrPDB, or $\semN$-\abbrPDB~\cite{DBLP:conf/pods/GreenKT07,feng:2019:sigmod:uncertainty}) are both formalized in \Cref{subsec:supp-mat-background}.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%END: move to appendix.
|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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We will use $\semNX$-\abbrPDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$ over the assignments to $\vct{X}$.
|
||||
We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-\abbrPDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{\numvar} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
|
||||
$\semNX$-\abbrPDB\xplural and a function $\rmod$ (which transforms an $\semNX$-\abbrPDB to a classical bag-\abbrPDB, or $\semN$-\abbrPDB~\cite{DBLP:conf/pods/GreenKT07,feng:2019:sigmod:uncertainty}) are both formalized in \Cref{subsec:supp-mat-background}.
|
||||
\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
|
||||
Given an $\semN$-\abbrPDB $\pdb = (\idb,\pd)$ and $\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, we have:
|
||||
$ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\randWorld\sim \pd'}\pbox{\polyForTuple(\randWorld)}. $
|
||||
Given an $\semN$-\abbrPDB $\pdb = (\idb,\pd)$ and equivalent polynomial $\polyForTuple$ for aribitrary tuple $\tup \in \pdb$,%$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
|
||||
we have:
|
||||
$ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\randWorld\sim \pd'}\pbox{\poly_{\query, \tup}(\randWorld)}. $
|
||||
\footnote{Although assumed by most prior work on set-probabilistic databases, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
|
||||
\end{Proposition}
|
||||
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.
|
||||
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 $\{0,1\}$.
|
||||
We focus on this problem from now on, assume an implicit result tuple, and so drop the subscript from $\polyForTuple$ (i.e., $\poly$ will denote a polynomial).
|
||||
We focus on this problem from now on, assume an implicit result tuple, and so drop the subscript from $\poly_{\query, \tup}$ (i.e., $\poly$ will denote a polynomial).
|
||||
|
||||
\subsubsection{\tis and \bis}
|
||||
\label{subsec:tidbs-and-bidbs}
|
||||
In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
|
||||
%
|
||||
A \bi $\pxdb = (\idb_{\semNX}, \pd)$ is an $\semNX$-\abbrPDB such that (i) every tuple is annotated with either $0$ (i.e., the tuple does not exist) 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 block are disjoint events.
|
||||
A \bi $\pdb$ is a \abbrPDB such that (i) every tuple is annotated with either $0$ (i.e., the tuple does not exist) or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pdb$ for which $\pdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same block are disjoint events.
|
||||
In other words, each random variable corresponds to the event of a single tuple's presence.
|
||||
%
|
||||
A \emph{\ti} is a \bi where each block contains exactly one tuple.
|
||||
\Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
|
||||
%
|
||||
In a \bi (and by extension a \ti) $\pxdb$, tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
|
||||
In a \bi (and by extension a \ti), tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
|
||||
Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function). For $t_{i, j} \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
|
||||
}
|
||||
Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
|
||||
Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ and the blocks induce the probability distribution $\pd$ of $\pdb$.
|
||||
We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
|
||||
$\poly(\vct{X})$ = $\poly(X_{1, 1},\ldots, X_{1, \abs{\block_1}},$ $\ldots, X_{\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$.\footnote{Later on in the paper, especially in \Cref{sec:algo}, we will overload notation and rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$.}
|
||||
|
||||
|
|
|
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Reference in New Issue