% Let $\poly(\vct{X})$ be a \bi-lineage polynomial.
% The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is the same as \Cref{def:reduced-poly} with the added constraint that all monomials with variables $X_{\block, i}, X_{\block, j}, i\neq j$ from the same block $\block$ are omitted.
%Consider a $\abbrBIDB$ polynomial $\poly\inparen{\vct{X}} = X_{1, 1}X_{1, 2} + X_{1, 2}X_{2, 1}^2$. Then by \Cref{def:reduced-bi-poly}, we have that $\rpoly\inparen{\vct{X}} = X_{1, 2}X_{2, 1}$. Next, we show why the reduced form is useful for our purposes.
%Let $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability distribution $\pdassign$ over $\{0,1\}^\numvar$ (the all worlds set) is induced by the probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$. As in \Cref{lem:tidb-reduce-poly} for \abbrTIDB, any \abbrBIDB-lineage polynomial $\poly(\vct{X})$ based on $\pdb$ and query $\query$ we have:
% % The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
If $\poly$ is a $1$-\abbrBIDB lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $\expct\pbox{\poly}=\rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $\bigO{\abs{\poly}}$ time.
\subsubsection{Possible World Semantics}\label{subsub:possible-world-sem}
Queries over probabilistic databases are traditionally viewed as being evaluated using the so-called possible world semantics. A general bag-\abbrPDB can be defined as the pair $\pdb=\inparen{\Omega, \bpd}$ where $\Omega$ is the set of possible worlds represented by $\pdb$. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\Omega$ is the set of query answers produced by evaluating $\query$ over each possible world $\omega\in\Omega$: $\inset{\query\inparen{\omega}: \omega\in\Omega}$.
The result of a query is the pair $\inparen{\query\inparen{\omega}, \bpd'}$ where $\bpd'$ is a probability distribution that assigns to each possible query result the sum of the probabilites of the worlds that produce this answer: $\probOf\pbox{\omega\in\Omega}=\sum_{\omega'\in\Omega,\\\query\inparen{\omega'}=\query\inparen{\omega}}\probOf\pbox{\omega'}$.
I am not sure the following paragraph is needed, since the reduction definition says pretty much the same thing. Unless that definition changes, we can get rid of this paragraph.
}
Suppose that $\pdb$ is a $1$-\abbrBIDB. Instead of looking only at the possible worlds of $\pdb$, one can consider all worlds, including those that cannot exist due to, e.g., disjointness. The all worlds set can be modeled by $\worldvec\in\{0, 1\}^{\bound\numvar}$, such that $\worldvec_{\tup, j}\in\worldvec$ represents whether or not the multiplicity of $\tup$ is $j$ (\emph{here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{\tup\in\tupset}\abs{b_\tup}$}).%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
We can denote a probability distribution over all $\worldvec\in\{0, 1\}^{\bound\numvar}$ as $\bpd'$. When $\bpd'$ is the one induced from each $\prob_{\tup, j}$ while assigning $\probOf\pbox{\worldvec}=0$ for any $\worldvec$ with $\worldvec_{\tup, j}, \worldvec_{\tup, j'}\geq1$ for $j\neq j'$, we end up with a bijective mapping from $\bpd$ to $\bpd'$, such that each mapping is equivalent, implying the distributions are equivalent.
\Cref{subsec:supp-mat-ti-bi-def} has more details.
Recall \Cref{fig:nxDBSemantics} again, which defines the lineage polynomial $\apolyqdt$ for any $\raPlus$ query. We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
Given a \abbrBPDB$\pdb=(\Omega,\bpd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for arbitrary result tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.\footnote{Although \Cref{prop:expection-of-polynom} follows, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
We focus on the problem of computing $\expct_{\worldvec\sim\pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ from now on, assume implicit $\query, \tupset, \tup$, and drop them from $\apolyqdt$ (i.e., $\poly\inparen{\vct{X}}$ will denote a polynomial).
