106 lines
10 KiB
TeX
106 lines
10 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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%\onecolumn
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\section{Background and Notation}\label{sec:background}
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\iffalse
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\subsection{Superlinearity of Bag \abbrPDB\xplural}\label{sec:suplin-bags}
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Moving forward, we focus exclusively on bags. For $Q()\dlImp$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$ $OnTime(\text{City}')$ over the bag relations of \Cref{fig:ex-shipping-simp}, consider the product query $\poly^2()\dlImp Q \times Q$.
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The factorized representation of $\poly^2$ is (for simplicity we ignore the random variables of $Route$ since each variable has probability of $1$):
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\begin{equation*}
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\poly^2 = \left(L_aL_b + L_bL_d + L_bL_c\right) \cdot \left(L_aL_b + L_bL_d + L_bL_c\right)
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\end{equation*}
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This equivalent SOP representation is
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\begin{equation*}
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L_a^2L_b^2 + L_b^2L_d^2 + L_b^2L_c^2 + 2L_aL_b^2L_d + 2L_aL_b^2L_c + 2L_b^2L_dL_c.
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\end{equation*}
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The expectation $\expct\pbox{\poly^2}$ then is:
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\begin{footnotesize}
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\begin{equation*}
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\expct\pbox{L_a^2}\expct\pbox{L_b^2} + \expct\pbox{L_b^2}\expct\pbox{L_d^2} + \expct\pbox{L_b^2}\expct\pbox{L_c^2} + 2\expct\pbox{L_a}\expct\pbox{L_b^2}\expct\pbox{L_d} + 2\expct\pbox{L_a}\expct\pbox{L_b^2}\expct\pbox{L_c} + 2\expct\pbox{L_b^2}\expct\pbox{L_d}\expct\pbox{L_c}
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\end{equation*}
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\end{footnotesize}
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Note that if $Dom(W_i) = \{0, 1\}$, then for any $k > 0$, $\expct\pbox{W_i^k} = \expct\pbox{W_i}$.
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This property leads us to consider a structure related to $\poly$.
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\begin{Definition}\label{def:reduced-poly}
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For any polynomial $\poly(\vct{X})$, define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in $\poly(\vct{X})$ to $1$.
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\end{Definition}
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With $\poly^2$ as an example, we have:
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\begin{align*}
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\rpoly^2(L_a, L_b, L_c, L_d)
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=&\; L_aL_b + L_bL_d + L_bW_c + 2L_aL_bL_d + 2L_aL_bL_c + 2L_bL_cL_d
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\end{align*}
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It can be verified that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(\probOf\pbox{L_a=1}, \probOf\pbox{L_b=1}, \probOf\pbox{L_c=1}), \probOf\pbox{L_d=1})$).
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The reduced form of a lineage polynomial can be obtained but requires a linear scan over the clauses of an SOP encoding of the polynomial. Note that for a compressed representation, this scheme would require an exponential number of computations in the size of the compressed representation. In \Cref{sec:hard}, we use $\rpoly$ to prove our hardness results .
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%In prior work on lineage-based Bag-\abbrPDB\xplural~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,yang:2015:pvldb:lenses} where this encoding is implicitly assumed, computing the expected count is linear in the size of the encoding.
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%In general however, compressed encodings of the polynomial can be exponentially smaller in $k$ for $k$-products --- the query $\poly^k$ obtained by taking the product of $k$ copies of $\poly$ as a factorized encoding of size $6\cdot k$, while the SOP encoding is of size $2\cdot 3^k$.
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%This leads us to the \textbf{central question of this paper}:
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%\begin{quote}
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%{\em
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%Is it always the case that the expectation of a UCQ in a Bag-\abbrPDB can be computed in time linear in the size of the \textbf{compressed} lineage polynomial?}
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%\end{quote}
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%If so, then Bag-\abbrPDB\xplural can indeed compete with deterministic databases.
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%This is unfortunately not the case, and an approximation is required.
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\fi
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\subsection{Probabilistic Databases (\abbrPDB\xplural)}
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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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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:
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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}. % and summarized here.
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$\semNX$-relations are functions from tuples to elements of $\semNX$, typically called annotations.
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We write $R(t)$ to denote the polynomial annotating tuple $t$ in relation $R$. Note that $R(t)$ is the lineage polynomial for $t$.
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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 $t \in R$ in this possible world, denoted $R(t)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $t$ on $\vct{\wElem}$.
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$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
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% For completeness, we briefly review the semantics for $\raPlus$ queries over $\semK$-relations~\cite{DBLP:conf/pods/GreenKT07}.
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% We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$. Below, we 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 $\pd$.
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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}.
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\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
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Given an $\semN$-\abbrPDB $\pdb = (\idb,\pd)$ and $\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, we have:
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$ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\randWorld\sim \pd'}\pbox{\polyForTuple(\randWorld)}. $
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\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.}
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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}.
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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 $\{0,1\}$.
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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).
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\subsubsection{\tis and \bis}
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\label{subsec:tidbs-and-bidbs}
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In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
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%
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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.
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In other words, each random variable corresponds to the event of a single tuple's presence.
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%
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A \emph{\ti} is a \bi where each block contains exactly one tuple.
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\Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
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%
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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{
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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 $t_j \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
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}
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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 $\pxdb$.
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We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
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$\poly(\vct{X})$ = $\poly(X_{1, 1},\ldots, X_{1, \abs{\block_1}},$ $\ldots, X_{\ell, \abs{\block_\ell}})$, 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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