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\subsection{Reduced Polynomials and Equivalences}

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.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{Definition}[Monomial]\label{def:monomial}
% A monomial is a product of a set of variables, each raised to a non-negative integer power.
% \end{Definition}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% For instance, the term $2xy$ contains a single monomial $xy$.
% \Cref{def:monomial} the monomial is $xy$.

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\begin{Definition}[Standard Monomial Basis]\label{def:smb}
A monomial is a product of variable terms, each raised to a non-negative integer power.
  A polynomial in \termSMB (\abbrSMB) has the form:
  \[
    \sum_{i=1}^n c_i \cdot m_i
  \]
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 $\smbOf{\poly}$.
%  fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
\end{Definition}
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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.

% \BG{Maybe inline degree?}
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\begin{Definition}[Degree]\label{def:degree}
The degree of polynomial $\poly(\vct{X})$ is the maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
\end{Definition}
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The degree of the running example polynomial is $2$. In this paper we consider only finite degree polynomials.
%
% Throughout this paper, we also make the following \textit{assumption}.
%
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% \begin{Assumption}\label{assump:poly-smb}
% 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 $\vct{d}$.
% \end{Assumption}
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%
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 $\pxdb$.
We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
$\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, 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 $X_{\block_1, 1}$ ?}
% and the probability distribution of $\pxdb$ is  uniquely determined based on a probability vector $\vct{p}$ that associates each tuple a probability
% 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 $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .





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\begin{definition}[Modding with a set]\label{def:mod-set}
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).
\end{definition}
For example when $S_0=\inset{X^2-X, Y^2-Y}$, taking the polynomial $2X^2 + 3XY - 2Y^2\mod S_0$ gives $2X+3XY-2Y$.
%
\begin{Definition}\label{def:mod-set-polys}
Given the set of BIDB variables $\inset{X_{b,i}}$, define
\[\mathcal{B}=\comprehension{X_{b,i}\cdot X_{b,j}}{\text{ for every block } b \text{and } i\ne j}\]
\[\mathcal{T}=\comprehension{X_{b,i}^2-X_{b,i}}{\text{ for every block } b \text{and } i}\]
\end{Definition}
%
\begin{Definition}[Reduced \bi Polynomials]\label{def:reduced-bi-poly}
  Let $\poly(\vct{X})$ be a \bi-lineage polynomial.
  The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is:
\begin{equation*}
\rpoly(\vct{X}) = \smbOf{\poly(\vct{X})} \mod \mathcal{T} \mod \mathcal{B}%X_i^2 - X_i \mod X_{\block_s, t}X_{\block_s, u}
\end{equation*}
%for all $i$ in $[\numvar]$ and for all $s$ in $\ell$, such that for all $t, u$ in $[\abs{\block_s}]$, $t \neq u$.
\end{Definition}
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%
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.
Alternatively, one can think of $\rpoly$ as the \abbrSMB of $\poly(\vct{X})$ when the product operator is idempotent.
%
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% \begin{Definition}[$\rpoly(\vct{X})$] \label{def:qtilde}
% Define $\rpoly(X_1,\ldots, X_\numvar)$ as the reduced version of $\poly(X_1,\ldots, X_\numvar)$, of the form
% $\rpoly(X_1,\ldots, X_\numvar) = $

% \[\poly(X_1,\ldots, X_\numvar) \mod X_1^2-X_1\cdots\mod X_\numvar^2 - X_\numvar.\]
% \end{Definition}
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\begin{Example}\label{example:qtilde}
Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blocks.  The expanded derivation for $\rpoly(X, Y)$ is
\begin{align*}
(&X^2 + 2XY + Y^2 \mod X^2 - X) \mod Y^2 - Y\\
= ~&X + 2XY + Y^2 \mod Y^2 - Y\\
= ~& X + 2XY + Y
\end{align*}
\end{Example}
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%
% 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 $e > 1$.
%
%When considering $\bi$ input, it becomes necessary to redefine $\rpoly(\vct{X})$.
%
\noindent The usefulness of this will reduction become clear in \Cref{lem:exp-poly-rpoly}.
%
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\begin{Definition}[Valid Worlds]
For probability distribution $\vct{P}$ and its corresponding PMF $P$, the set of valid worlds $\eta$ corresponds to all worlds that have a probability value greater than $0$, formally, for random variable $\vct{W}$
\[
\eta = \{\vct{w}\st P[\vct{W} = \vct{w}] > 0\}
\]
\end{Definition}

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}.
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%Define all variables $X_i$ in $\poly$ to be independent.

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\begin{Lemma}\label{lem:exp-poly-rpoly}
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 some query $\query$ we have
  % The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
\begin{equation*}
\expct_{\vct{w}\sim \vct{p}}\pbox{\poly(\vct{W})}  = \rpoly(\vct{p}).
\end{equation*}
\end{Lemma}
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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.



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\begin{Corollary}\label{cor:expct-sop}
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 $\poly$.
\end{Corollary}
%\AH{What if $\poly$ is not in \abbrSMB form?}


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