%\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}
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 $\db_j$.
Denote the schema of $\db$ as $\sch(\db)$. 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 a 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 defined as the set of results (worlds) from running $\query$ over each $\db_i \in\idb$. We write this formally as,
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 \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.
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.
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 annotating tuple $\tup$.
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.
The above semantics show us how to obtain the annotation on a tuple in the result of query $\query$ from the annotations on the tuples in the input of $\query$.
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. 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 $\{0, 1\}^M$ is 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 $\llbracket\poly(\db)\rrbracket(\tup)$, where the injective mapping maps $\db$ to $\vct{X}$.
\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?}
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 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,