76 lines
7.9 KiB
TeX
76 lines
7.9 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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\onecolumn
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\section{Query translation into polynomials}
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%\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}
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%1) RA notation
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%2) DB (TIDB) notation
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%3) How queries translate into polynomials
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%}
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\subsection{Introduction}
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An incomplete database $\idb$ is a set of deterministic databases $\db$ where each element is known as a possible world. %Since $\idb$ is modeling all the possible worlds of an uncertain database, it follows that each $\db \in \idb$ has the same named set of relations, $\{\rel_1,\ldots, \rel_n\}$ (albeit not equivalent across all instances), whose schemas $(\sch(\rel_i))$are unchanging across each $\db_j$.
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Denote the schema of $\db$ as $\sch(\db)$. When $\idb$ is a probabilistic database, $\idb$ can be viewed as a two tuple $(\wSet, \pd)$, where $\wSet$ as noted, is the set of possible worlds, and $\pd$ is a probability distribution over $\wSet$.
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The possible worlds semantics gives a framework for how to think about running queries over $\idb$. Given a query $\query$, $\query$ is deterministically run over each $\db \in \idb$, and the output of $\query(\idb)$ is defined as the set of results (worlds) from running $\query$ over each $\db_i \in \idb$. We write this formally as,
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\[\query(\idb) = \{\query(\db) | \db \in \idb\}\]
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\subsection{Modeling and Semantics}
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Define $\vct{X}$ to be the variables $X_1,\dots,X_M$. We emphasize that formal variables do not have a fixed domain type prior to assignment. Let the set of all tuples in domain of $\sch(\db)$ be $\tset$.
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\subsubsection{K-relations}\label{subsubsec:k-rel}
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A K-relation~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are each annotated with an expression whose values come from its respective commutative K-semiring, denoted $\{K, \oplus, \otimes, \mathbbold{0}, \mathbbold{1}\}$. A commutative $K$-semiring has associative and commutative operators $\oplus$ and $\otimes$, with $\otimes$ distributing over $\oplus$, $\mathbbold{0}$ the identity of $\oplus$, $\mathbbold{1}$ likewise of $\otimes$, and element $\mathbbold{0}$ anihilates all elements of $K$ when being combined with $\otimes$. The information encoded in the annotation depends on the underlying semiring of the relation.
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As noted in \cite{DBLP:conf/pods/GreenKT07}, the $\mathbb{N}[\vct{X}]$-semiring is a semiring over the set $\mathbb{N}[\vct{X}]$ of all polynomials, whose variables can then be substituted with $K$-values from other semirings, evaluating the operators with the operators of the substituted semiring, to produce varying semantics such as set, bag, and security annotations.
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When used with $\mathbb B$-typed variables, an $\mathbb{N}[\vct{X}]$ relation is effectively a C-Table \cite{DBLP:conf/pods/GreenKT07}, since all first order formulas can be equivalently modeled by polynomials, where disjunction is equivalent to the addition operator and conjunction is equivalent to the multiplication operator.
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Using $\mathbb B$-typed variables in an $\mathbb{N}[\vct{X}]$ relation would correspond to substituting values and operators from the $\{\mathbb{B}, \vee, \wedge, \bot, \top\}$ semiring.
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Further define $\nxdb$ as an $\mathbb{N}[\vct{X}]$ database where each tuple $\tup \in \db$ is annotated with a polynomial over variables $X_1,\ldots, X_M$ for some value of $M$ that will be specified later.
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Since $\nxdb$ is a database that maps tuples to polynomials, it is customary for arbitrary table $\rel$ to be viewed as a function $\rel: \tset \mapsto \mathbb{N}[\vct{X}]$, where $\rel(\tup)$ denotes the polynomial annotating tuple $\tup$.
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It has been shown in previous work that commutative semirings precisely model translations of RA+ query operations to $k$-annotations.
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The evalution semantics notation $\llbracket \cdot \rrbracket = x$ simply mean that the result of evaluating expression $\cdot$ is given by following the semantics $x$. Given a query $\query$, operations in $\query$ are translated into the following polynomial expressions.
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\begin{align*}
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&\eval{\project_A(\rel)}(\tup)&& = &&\sum_{\tup': \project_A(\tup) = \tup} \eval{\rel}(\tup')\\
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&\eval{(\rel_1 \union \rel_2)}(\tup)&& = &&\eval{\rel_1}(\tup) + \eval{\rel_2}(\tup)\\
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&\eval{(\rel_1 \join \rel_2)}(\tup) && = &&\eval{\rel_1}(\project_{\sch(\rel_1)}(\tup)) \times \eval{\rel_2}(\project_{\sch(\rel_2)}(\tup)) \\
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&\eval{\select_\theta(\rel)}(\tup) && = &&\begin{cases}
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\eval{\rel}(\tup) &\text{if }\theta(\tup) = 1\\
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0 &\text{otherwise}.
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\end{cases}\\
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&\eval{R}(\tup) && = &&\rel(\tup)
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\end{align*}
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The above semantics show us how to obtain the $k$-annotation on a tuple in the result of query $\query$ from the annotations on the tuples in the input of $\query$.
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\subsection{Defining the Data}\label{subsec:def-data}
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For the set of possible worlds, $\wSet$, i.e. the set of all $\db_i \in \idb$, define an injective mapping to the set $\{0, 1\}^M$, where for each vector $\vct{w} \in \{0, 1\}^M$ there is at most one element $\db_i \in \idb$ mapped to $\vct{w}$.
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In the general case, the binary value of $\vct{w}$ uniquely identifies a potential possible world. For example, consider the case of the Tuple Independent Database $(\ti)$ data model in which each table is a set of tuples, each of which is independent of one another, and individually occur with a specific probability $\prob_\tup$. Because of independence, a $\ti$ with $\numTup$ tuples naturally has $2^\numTup$ possible worlds, thus $\numTup = M$, and the injective mapping for each $\vct{w} \in \{0, 1\}^M$ is trivial. In the Block Independent Disjoint data model ($\bi$), because of the disjoint condition on tuples within the same block, a $\bi$ may not have exactly $2^M$ possible worlds. Excess $\vct{w}$'s are assigned a probability of $0$.
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Denote a random variable selecting a world according to distribution $P$ to be $\rw$. Provided that for any non-possible world $\vct{w} \in \{0, 1\}^M, \pd[\rw = \vct{w}] = 0$, a probability distribution over $\{0, 1\}^M$ is a distribution over $\Omega$, which we have already defined as $\pd$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%This could be a way to think of world binary vectors in the general case
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%Let $\vct{w}$ be a $\left\lceil\log_2\left(\left|\wSet\right|\right)\right\rceil = \numTup$ binary bit vector, uniquely identifying possible world $\db_i \in \idb$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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From this point on our discussion focuses on exactly one specific tuple $\tup$. Thus, we abuse notation by using $\poly(\vct{X})$ to be the annotated polynomial $\llbracket\poly(\db)\rrbracket(\tup)$, and for a domain of $\{0, 1\}$ for each $X_i \in \vct{X}$, the injective mapping maps $\db$ to $\vct{X}$.
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One of the aggregates we desire to compute over the annotated polynomial is the expectation over possible worlds, denoted,
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\[\expct_{\vct{\rw} \sim \pd}\pbox{\poly(\rw)} = \sum\limits_{\wVec \in \{0, 1\}^\numTup} \poly(\wVec)\cdot \pd[\rw = \vct{w}].\]
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Above, $\poly(\vct{w})$ is used to mean the assignment of $\vct{w}$ to $\vct{X}$.
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For a $\ti$, the bit-string world value $\vct{w}$ can be used as indexing to determine which tuples are present in the $\vct{w}$ world, where the $i^{th}$ bit position $(\wbit_i)$ represents whether a tuple $\tup_i$ appears in the unique world identified by the binary value of $\vct{w}[i]$. Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$ such that those probabilities produce the possible worlds in D with a distribution $\pd$ over all worlds. Let $\pd^{(\vct{p})}$ represent the distribution induced by $\vct{p}$.
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\[\expct_{\rw\sim \pd^{(\vct{p})}}\pbox{\poly(\rw)} = \sum\limits_{\wVec \in \{0, 1\}^\numTup} \poly(\wVec)\prod_{\substack{i \in [\numTup]\\ s.t. \wElem_i = 1}}\prob_i \prod_{\substack{i \in [\numTup]\\s.t. w_i = 0}}\left(1 - \prob_i\right).\]
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