63 lines
6.4 KiB
TeX
63 lines
6.4 KiB
TeX
%root: main.tex
|
|
%!TEX root=./main.tex
|
|
%\onecolumn
|
|
\section{Background and Notation}\label{sec:background}
|
|
|
|
\subsection{Probabilistic Databases}
|
|
|
|
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.
|
|
|
|
An \textit{incomplete database} $\idb$ is a set of deterministic databases $\db$ called possible worlds.
|
|
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: $\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}$.
|
|
|
|
For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$ that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
|
|
%
|
|
\[\forall \db \in \query(\idb): \pd'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \pd(\db') \]
|
|
|
|
Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
|
|
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}.
|
|
$\semNX$-relations are functions from tuples to elements of $\semNX$, typically called annotations.
|
|
We write $R(t)$ to denote the polynomial annotating tuple $t$ in relation $R$. Note that $R(t)$ is the lineage polynomial for $t$.
|
|
Each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
|
|
The multiplicity of $t \in R$ in this possible world, denoted $R(t)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $t$ on $\vct{\wElem}$.
|
|
$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
We will use $\semNX$-\abbrPDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$.
|
|
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)}. $
|
|
\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).
|
|
|
|
\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.
|
|
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{
|
|
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 $\pxdb$.
|
|
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 $n=\sum_{i=1}^\ell \abs{b_i}$.}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
%%% Local Variables:
|
|
%%% mode: latex
|
|
%%% TeX-master: "main"
|
|
%%% End:
|