We now introduce some terminology for polynomials and develop a reduced form for polynomials --- a closed form of the polynomial's expectation over probability distributions derived from a \bi or \ti.
Throughout, we will use $(X + Y)^2$ as a running example.
where each $c_i$ is a positive integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. The \abbrSMB of a polynomial $\poly$ is $\smbOf{\poly}$.
The \abbrSMB for the running example is $X^2+2XY + Y^2$. While $X^2+ XY + XY + Y^2$ is an expanded form of the expression, it is not the standard monomial basis since $XY$ appears more than once.
% All polynomials considered are in standard monomial basis, i.e., $\poly(\vct{X}) = \sum\limits_{\vct{d} \in \mathbb{N}^\numvar}q_d \cdot \prod\limits_{i = 1, d_i \geq 1}^{\numvar}X_i^{d_i}$, where $q_d$ is the coefficient for the monomial encoded in $\vct{d}$ and $d_i$ is the $i^{th}$ element of $\vct{d}$.
We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if
%\AH{Why is it required for the tuple to be n-ary? I think this slightly confuses me since we have n tuples.}
% OK: agreed w/ AH, this can be treated as implicit
there exists a $\raPlus$ query $\query$, \bi$\pxdb$ (\ti$\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\query(\vct{X})=\query(\pxdb)(\tup)$. % Before proceeding, note that the following is assume that polynomials are \bis (which subsume \tis as a special case).
As they are a special case of \bis, the following applies to \tis as well.
Recall that in a \bi$\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$.
Tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_i$ is associated with a probability $\prob(\tup_i)=\pd[X_i =1]$.\footnote{
Note the deviation from the more common approach of defining a single independent, $[\abs{\block_i}+1]$-valued variable per block; Here we define $\abs{\block_i}$ correlated variables per block.
}
Because blocks are independent and tuples from the same block are disjoint, $\prob$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
$\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$$\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $X_{i, j}$ denotes the annotation of tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
%\SF{Where is $\block_{i, j}$ used? Is it $X_{\block_{1, 1}}$ or $X_{\block_1, 1}$ ?}
% variables are independent of each other (or disjoint if they are from the same block) and each variable $X$ is associated with a probability $\vct{p}(X) = \pd[X = 1]$. Thus, we are dealing with polynomials $\poly(\vct{X})$ that are annotations of a tuple in the result of a query $\query$ over a BIDB $\pxdb$ where $\vct{X}$ is the set of variables that occur in annotations of tuples of $\pxdb$.
% While the definition of polynomial $\poly(\vct{X})$ over a $\bi$ input doesn't change, we introduce an alternative notation which will come in handy. Given $\ell$ blocks, we write $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
% The number of tuples in the $\bi$ instance can be (trivially) computed as $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .
Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
Intuitively, in the reduced form, all exponents $e > 1$ are reduced to $e =1$ and all monomials with multile variables from the same block $\block$ are dropped (any world containing more than one tuple from a block has $0$ probability and can be ignored).
For the special case of \tis, the second step is not necessary since every block contains a single tuple.
% Intuitively, $\rpoly(\textbf{X})$ is the \abbrSMB form of $\poly(\textbf{X})$ such that if any $X_j$ term has an exponent $e > 1$, it is reduced to $1$, i.e. $X_j^e\mapsto X_j$ for any $e > 1$.
For probability distribution $\vct{P}$ and its corresponding PMF $P$, the set of valid worlds $\eta$ is the worlds with probability value greater than $0$; i.e., for variable vector $\vct{W}$
We state additional equivalences between $\poly(\vct{X})$ and $\rpoly(\vct{X})$ in~\Cref{app:subsec-pre-poly-rpoly} and~\Cref{app:subsec-prop-q-qtilde}.
Let $\pxdb$ be a \bi over variables $\vct{X}=\{X_1, \ldots, X_\numvar\}$ and with probability distribution $\vct{p}=(\prob_1, \ldots, \prob_\numvar)$ over all $\vct{w}$ in $\eta$. For any \bi-lineage polynomial $\poly(\vct{X})$ based on $\pxdb$ and query $\query$ we have:
Note that in the preceding lemma, we have assigned $\vct{p}$
%(introduced in \Cref{subsec:def-data})
to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that when we replace each variable $X_i$ with its probability $\prob_i$ in the reduced form of a \bi-lineage polynomial and evaluate the resulting expression in $\mathbb{R}$, then the result is the expectation of the polynomial.
If $\poly$ is a \bi-lineage polynomial, then the expectation of $\poly$, i.e., $\expct\pbox{\poly}=\rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $O(|\smbOf{\poly}|)$, where $|\poly|$ denotes the total number of multiplication/addition operators in $\poly$.