%root: main.tex
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\onecolumn
\section{Query translation into polynomials}
%\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}
%1) RA notation
%2) DB (TIDB) notation
%3) How queries translate into polynomials
%}
\subsection{Introduction}


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 $\db_j$.  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 $\vct{w}$.  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 the probability distribution over $\wSet$.  

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 the set of results (worlds) from running $\query$ over each $\db_i \in \idb$.  We write this formally as,
\[\query(\idb) = \{\query(\db) | \db \in \idb\}\]



\subsection{Modeling and Semantics}
Define $\vct{X}$ to be the variables $X_1,\dots,X_M$.  Let the set of all tuples in domain $\mathbb{D}$ be $\tset$.


\subsubsection{K-relations}\label{subsubsec:k-rel}
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.  
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.  

When used with $\mathbb B$-typed variables, an $\mathbb{N}[\vct{X}]$ relation is effectively a C-Table, 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, and in boolean semantics, negation of variable $x$ can be easily translated into $(1 - x)$.

This would correspond to substituting values and operators from the $\{\mathbb{B}, \vee, \wedge, \bot, \top\}$ semiring.

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.  
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 mapped to tuple $\tup$.

It has been shown in previous work that commutative semirings precisely model translations of RA+ query operations to set annotations.  
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 operations.

\begin{align*}
&\eval{\project_A(\rel)}(\tup) = &&\sum_{\tup': \project_A(\tup) = \tup} \eval{\rel}(\tup')\\
&\eval{(\rel_1 \union \rel_2)}(\tup) = &&\eval{\rel_1}(\tup) + \eval{\rel_2}(\tup)\\
&\eval{(\rel_1 \join \rel_2)}(\tup) = &&\eval{\rel_1}(\project_{\sch(\rel_1)}(\tup)) \times \eval{\rel_2}(\project_{\sch(\rel_2)}(\tup))	\\
&\eval{\select_\theta(\rel)}(\tup) = &&\begin{cases}
					\eval{\rel}(\tup)	&\text{if }\theta(\tup) = 1\\
					0		&\text{otherwise}.
				\end{cases}\\
&\eval{R}(\tup) = &&\rel(\tup)
\end{align*}

The above semantics show us how to obtain the annotation on a tuple in the result of q query $\query$ from the annotations on the tuples in the input of $\query$.

\subsection{Defining the Data}

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 are 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 each $\vct{w} \in \{0, 1\}^M$ is indeed a possible world.  However in the Block Independent Disjoint data model (BIDB), because of the disjoint condition on tuples within the same block, a BIDB may not have exactly $2^M$ possible worlds. Such $\vct{w}$'s, that do not exist, are assigned a probability of $0$.  

Denote a random variable selecting a world according to distribution $P$ to be $\rw$.  Provided that for any non-possible world $\vct{w} \in \{0, 1\}^M, \pd[\rw = \vct{w}] = 0$, then, a probability distribution over $\{0, 1\}^M$ implies a distribution over $\Omega$, which we have already defined as $\pd$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This could be a way to think of world binary vectors in the general case
%Let $\vct{w}$ be a $\left\lceil\log_2\left(\left|\wSet\right|\right)\right\rceil = \numTup$ binary bit vector, uniquely identifying possible world $\db_i \in \idb$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Assume a domain of $\{0, 1\}$ for each $X_i \in \vct{X}$.  Since, from this point on, our discussion will involve one polynomial for an arbirtrary $\tup$, we thus abuse notation by using $\poly(\vct{X})$ to be the annotated polynomial $\llbracket\poly(\idb)\rrbracket(\tup)$.  

One of the aggregates we desire to compute over the annotated polynomial is the expectation, denoted,

\AH{With our notation, I no longer think that $\vct{w} \sim \pd$ is necessary footer for $\expct$.  We can probably just have $\expct\limits_{\vct{w}}$ instead.  Do you agree?}
\AR{No. How would you state Lemma 4 without explicitly using $P$ in the definition of expectation?}

\[\expct_{\vct{\rw} \sim \pd}\pbox{\poly(\rw)} = \sum\limits_{\wVec \in \{0, 1\}^\numTup} \poly(\wVec)\cdot \pd[\rw = \vct{w}].\]

If there are exactly $2^M$ possible worlds (e.g., as in 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 represents whether a tuple $\tup_i$ appears in the unique world identified by the binary value of $\vct{w}$.  Given an $\numTup$-sized vector $\vct{p}$, where the $i^{th}$ element, $\prob_i$ is the probability of the $i^{th}$ tuple, denote the vector $\vct{p}$ according to the probability distributation $\pd$ as $\pd^{(\vct{p})}$.  We can then write an equivalent expectation for $\ti$ model,

\[\expct_{\rw\sim \pd^{(\vct{p})}}\pbox{\poly(\rw)} = \sum\limits_{\wVec \in \{0, 1\}^\numTup} \poly(\wVec)\prod_{\substack{i \in [\numTup]\\ s.t. \wElem_i = 1}}\prob_i \prod_{\substack{i \in [\numTup]\\s.t. w_i = 0}}\left(1 - \prob_i\right).\]
