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.
\AH{Reduction \emph{I think} combined with~\Cref{lem:tidb-reduce-poly} should replace this. In fact~\Cref{lem:tidb-reduce-poly} works for also for \abbrBIDB\xplural, so, maybe still stating this, but that it follows from~\Cref{lem:tidb-reduce-poly}.}
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:
If $\poly$ is a \bi-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.