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\subsection{Reduced Polynomials and Equivalences}

We now introduce some terminology % for polynomials
and develop a reduced form of lineage polynomials for a \abbrBIDB or \abbrTIDB.
%We will use $(X + Y)^2$ as a running example.
Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ with individual degree $B <\infty$ 
%\footnote{The standard definition of polynomials requires a finite number of terms.} and $c_\vct{i} \in \domN$ 
is formally defined as: %(with $c_\vct{i} \in \domN$):
\AH{Do we want to say that $\domain\inparen{c_\vct{i}} = \domR$ instead?  I've only seen the $\semNX$ use case.  Is there a legitimate use case for real valued coefficients?}
\begin{equation}
  \label{eq:sop-form}
\poly\inparen{X_1,\dots,X_n}=\sum_{\vct{d}\in\{0,\ldots,B\}^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i},
\end{equation}
where $c_{\vct{d}}\in \semN$.

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\begin{Definition}[Standard Monomial Basis]\label{def:smb}
From above, the term $\prod_{i=1}^n X_i^{d_i}$ is a {\em monomial}. A polynomial $\poly\inparen{\vct{X}}$ is in standard monomial basis (\abbrSMB) when we keep only the terms with $c_{\vct{d}}\ne 0$ from \Cref{eq:sop-form}.
\end{Definition}
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Unless othewise noted, we consider all polynomials to be in \abbrSMB representation. 
When it is unclear, we use $\smbOf{\poly}$ to denote the \abbrSMB form of a polynomial $\poly$.

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\begin{Definition}[Degree]\label{def:degree}
The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such that $c_{(d_1,\dots,d_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
\end{Definition}
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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 to produce an output tuple.
%in any clause of the $\raPlus$ query that created it.
We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \bi (\ti) $\pdb$, and tuple $\tup$ such that $\poly\inparen{\vct{X}} = \query(\pdb)(\tup)$. 

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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 for a set of polynomials $S=\inset{X^2-X, Y^2-Y}$, taking the polynomial $2X^2 + 3XY - 2Y^2\mod S$ yields $2X+3XY-2Y$.
%%
%\begin{Definition}[$\mathcal B$, $\mathcal T$]\label{def:mod-set-polys}
%Given the set of BIDB variables $\inset{X_{i,j}}$, define
%
%\setlength\parindent{0pt}
%\vspace*{-3mm}
%{\small
%\begin{tabular}{@{}l l}
%	\begin{minipage}[b]{0.45\linewidth}
%		\centering
%		\begin{equation*}
%		\mathcal{B}=\comprehension{X_{i,j}\cdot X_{i,j'}}{i \in [\ell], j\neq j' \in [~\abs{\block_i}~]}
%		\end{equation*}
%	\end{minipage}%
%	\hspace{13mm}
%	&
%	\begin{minipage}[b]{0.45\linewidth}
%		\centering
%		\begin{equation*}
%		\mathcal{T}=\comprehension{X_{i,j}^2-X_{i,j}}{i \in [\ell], j \in [~\abs{\block_i}~]}
%		\end{equation*}
%	\end{minipage}
%	\\
%\end{tabular}
%}
%\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 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.
  
%: $\rpoly(\vct{X}) = \poly(\vct{X}) \mod \inparen{\mathcal{T} \cup \mathcal{B}}$
\end{Definition}
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%

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.
%, (recall the constraint on tuples from the same block being disjoint in a \bi).% any monomial 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.
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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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%%Removing this example to save space
\iffalse
\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}
\fi
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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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%NEEDS to be moved to the appendix
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%\begin{Definition}[Valid Worlds]
%For probability distribution $\pd$, % and its corresponding probability mass function $\probOf$,
%the set of valid worlds $\valworlds$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
%\[
%\valworlds = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
%\]
%\end{Definition}
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%END move to appendix
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%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 $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability distribution $\pdassign$ over $\vct{\randWorld}^\numvar$ (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 $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
\begin{equation*}
\expct_{\vct{W}\sim \pd}\pbox{\poly(\vct{W})}  = \rpoly(\probAllTup).
\end{equation*}
\end{Lemma}
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By \Cref{lem:exp-poly-rpoly} and linearity of expectation, the following corollary results.
%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 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 $\bigO{\size\inparen{\poly}}$, where $\size\inparen{\poly}$ (\Cref{def:size}) is proportional to the total number of multiplication/addition operators in $\poly$.
\end{Corollary}


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