\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}
Given tables $\rel, \reli$, an arbitrary query $\query(\rel)$ over the positive relational operators (SPJU), abusing notation slightly denote the query polynomial as $\poly(X_1,\ldots, X_\numTup)$. To be clear, $\poly(X_1,\ldots, X_\numTup)$ is a polynomial whose variables represent the tuple annotations of an arbitrary query.The annotation for arbitrary tuple $\tup$ can be viewed as an element of the image of $\rel$, where relation $\rel$ can be thought of as a function with preimage of all tuples in $\rel$, such that $\rel(\tup)=\poly(X_1,\ldots, X_\numTup)$. Further, it is known that the algebraic semiring structure aptly models the translation and computation of query operations into tuple annotation, aka polynomials.
To make things more concrete, consider the $\{\mathbb{N}, \times, +, 1, 0\}$ bag semiring. Here the set in which the tuple annotations (computed polynomials) exist is the natural numbers. Query operations are translated into one of the two semiring operators, with $\project$ and $\union$ of agreeing tuples being the equivalent of the '+' opertator in polynomial $\poly$, $\join$ translating into the $\times$ operator, and finally, $\select$ is better modeled as a function that returns either $\rel(\tup)$ or $0$ based on some predicate.
Consider the translation of relational operators to polynomial operators in greater detail.
A specific probabilistic data model is the Tuple Independent Database (\ti). This is a database 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, $\prob_\tup$.
There are features of $\ti$ that we can exploit. Note that a $\ti$ naturally has $2^\numTup$ possible worlds, each of which can be conveniently modeled by an $\numTup$ bit string. The bit-string world value can be used as an index to determine which tuples are present in the $\wVec$ world. We can then write and equivalent expectation for $\ti$ models,