27 lines
3.8 KiB
TeX
27 lines
3.8 KiB
TeX
%!TEX root=./main.tex
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We use $\domK$-relations to model bags. A \emph{$\domK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$. A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
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Let $\udom$ be a countable domain of values.
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Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$.
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A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.
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For completeness, we briefly review the semantics for $\raPlus$ queries over $\semK$-relations~\cite{DBLP:conf/pods/GreenKT07} illustrated in \Cref{fig:nxDBSemantics}.
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In \Cref{fig:nxDBSemantics}, we use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$, assume that tuples are of appropriate arity, use $\sch(\rel)$ to denote the attributes of $\rel$, and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$. Furthermore, $\theta(\tup)$ denotes the (Boolean) result of evaluating condition $\theta$ over $\tup$.
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Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result tuple $t$, we use $\query(\db)(t)$ for $\db \in \idb$ as input in RHS of \Cref{eq:intro-bag-expectation} to compute the expected multiplicity of $t$. Note that the tables of \Cref{fig:ex-shipping-simp} have an implicit $1$ $\semN$-valued annotation for each tuple in tables $OnTime$ and $Route$.
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%\Cref{ex:intro-tbls} and \Cref{ex:intro-lineage} $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \probDist}[\query(\db)(t)]$:
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%%
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%\begin{equation}\label{eq:bag-expectation}
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%\expct_{\idb \sim \probDist}[\query(\db)(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \probOf(\db)
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%\end{equation}
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%%
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Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of $t$ we expect to find in result of query $\query$.
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Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_n)$ with natural number coefficients and exponents.
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Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$, with the standard addition and multiplication of polynomials.
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We will use $\semNX$-PDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$.
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We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-PDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{|\vct X|} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
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$\semNX$-PDBs and a function $\rmod$ (which transforms an $\semNX$-PDB to an equivalent $\semN$-PDB) are both formalized next.
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