Given an index set $S$ over variables $X_\tup$ for $\tup\in S$, a (general) polynomial $\genpoly$ over $\inparen{X_\tup}_{\tup\in S}$ with individual degree $\hideg <\infty$
The term $\prod_{\tup\in S} X_\tup^{d_\tup}$ in \Cref{eq:sop-form} is a {\em monomial}. A polynomial $\genpoly\inparen{\vct{X}}$ is in standard monomial basis (\abbrSMB) when we keep only the terms with $c_{\vct{d}}\ne0$ from \Cref{eq:sop-form}.
When it is unclear, we use $\smbOf{\genpoly}~\inparen{\smbOf{\poly}}$ to denote the \abbrSMB form of a polynomial (lineage polynomial) $\genpoly~\inparen{\poly}$.
As an example, the degree of the polynomial $X^2+2XY^2+Y^2$ is $3$.
Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins needed to produce a result tuple.
or simply lineage polynomial), if it is clear from context that there exists an $\raPlus$ query $\query$, \abbrCTIDB$\pdb$, and result tuple $\tup$ such that $\poly\inparen{\vct{X}}=\apolyqdt\inparen{\vct{X}}.$
\noindent A block independent database \abbrBIDB$\pdb'$ models a set of worlds each of which consists of a subset of the possible tuples $\tupset'$, where $\tupset'$ is partitioned into $\numblock$ blocks $\block_i$ and all $\block_i$ are independent random events. $\pdb'$ further constrains that all $\tup\in\block_i$ for all $i\in\pbox{\numblock}$ of $\tupset'$ be disjoint events. We refer to any monomial that includes $X_\tup X_{\tup'}$ for $\tup\neq\tup'\in\block_i$ as a \emph{cancellation}. We define next a specific construction of \abbrBIDB that is useful for our work.
Define a \emph{\abbrOneBIDB} to be the pair $\pdb' =\inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$ where $\tupset'$ is the set of possible tuples such that each $\tup\in\tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$. $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples. $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup\in\block_i}\prob_\tup\leq1$. Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_i(W)=\begin{cases}
\noindent$\bpd'$ is the probability distribution across all worlds such that, given $W\in\bigtimes_{\tup\in\tupset'}\inset{0,\bound_\tup}$, $\probOf\pbox{\worldvec= W}=\prod_{i\in\pbox{\numblock}}\prob_{i}(W)$.
We slightly abuse notation here, denoting a world vector as $W$ rather than $\worldvec$ to distinguish between the random variable and the world instance. When there is no ambiguity, we will denote a world vector as $\worldvec$.}
\Cref{fig:lin-poly-bidb} shows the lineage construction of $\poly'\inparen{\vct{X}}$ given $\raPlus$ query $\query$ for arbitrary deterministic $\gentupset'$. Note that the semantics differ from~\Cref{fig:nxDBSemantics} only in the base case.
Given \abbrCTIDB$\pdb=\inparen{\worlds, \bpd}$, let $\pdb' =\inparen{\onebidbworlds{\tupset'}, \bpd'}$ be the \emph{\abbrOneBIDB} obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup=\inset{\intup{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$.
The probability distribution $\bpd'$ is the characterized by the vector $\vct{p}=\inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$.
Given a polynomial $\poly'\inparen{\vct{X}}$ generated from a \abbrOneBIDB and let $\rpoly'\inparen{\vct{X}}$ denote the reduced form of $\poly'\inparen{\vct{X}}$ derived as follows: i) compute $\smbOf{\poly'\inparen{\vct{X}}}$ eliminating all monomials with cross terms $X_{\tup}X_{\tup'}$ for $\tup\neq\tup' \in\block_i$ and ii) reduce all \emph{variable} exponents $e > 1$ to $1$.
Then given $\worldvec\in\inset{0,1}^{\tupset'}$ over the reduced \abbrOneBIDB of~\Cref{prop:ctidb-reduct}, the disjoint requirement and the semantics for constructing the lineage polynomial over a \abbrOneBIDB, $\poly'\inparen{\worldvec}$ is of the same structure as the reformulated polynomial $\refpoly{}\inparen{\worldvec}$ of step i) from~\Cref{def:reduced-poly}, which then implies that $\rpoly'$ is the reduced polynomial that results from step ii) of both~\Cref{def:reduced-poly} and~\Cref{def:reduced-poly-one-bidb}, and further that~\Cref{lem:tidb-reduce-poly} immediately follows for \abbrOneBIDB polynomials.