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\subsection{Reduced Polynomials and Equivalences}

Since we have shown that computing the expected multiplicity of a query result tuple is equivalent to computing the expectation of a polynomial (for that tuple) given a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$, we from now on focus on this problem exclusively.
We now introduce some basic terminology for polynomials and then develop a reduced normal form for polynomials that preserves a polynomial expectation for probability distributions that stems from \bis or \tis.
Let us use the expression $(x + y)^2$ as a running example in this section.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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 $xy$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[Standard Monomial Basis]\label{def:smb}
A monomial is a product of a set of variables, each raised to a non-negative integer power.
  A polynomial is in \termSMB (\abbrSMB) when it is of 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 $i \neq j$. Given a polynomial $\poly$ we denote its \abbrSMB as $\smbOf{\poly}$.
%  fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
\end{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[Degree]\label{def:degree}
The degree of polynomial $\poly(\vct{X})$ is the maximum sum of the exponents of a monomial, over all monomials in $\smbOf{\poly(\vct{X})}$.
\end{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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}.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (\emph{\ti-lineage polynomial}), if there exists an n-ary $\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$), and n-ary 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).
Note the \tis are a special case of \bis and, thus, the following applies to \tis as well.
Recall that in a \bi $\pdbx$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pdbx$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\vct{p}(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\vct{p}$ and the blocks induce a the probability distribution $\pd$ of $\pdbx$.
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 $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
% and the probability distribution of $\pdbx$ 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 $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 defined as
\begin{equation*}
\rpoly(\vct{X}) = \smbOf{\poly(\vct{X})} \mod 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Intuitively, in the reduced form all exponents $e > 1$ are reduced to $e = 1$ and, all monomials containing more than one variable from the same block $\block$ are dropped. Note that for the special case of \tis, there is no dropping of monomials 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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Example}\label{example:qtilde}
Consider $\poly(x, y) = (x + y)(x + y)$ where $x$ and $y$ are from different blocks.  Then 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 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 $\rpoly(\vct{X})$.

The usefulness of this reduction become clear in \Cref{lem:exp-poly-rpoly}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Lemma}\label{lem:pre-poly-rpoly}
When $\poly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}} \cdot \prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i^{d_i}$, we have then that $\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar} q_{\vct{d}}\cdot\prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i$.
\end{Lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Follows by the construction of $\rpoly$ in \cref{def:reduced-bi-poly}. \qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Note the following fact:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{w} \in \{0,1\}^\numvar$,
  \[
    \poly(\vct{w}) = \rpoly(\vct{w}).
    \]
\end{Proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof for Proposition ~\ref{proposition:q-qtilde}]
Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion.  For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$. \qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $\vct{p} = (\prob_1, \ldots, \prob_\numvar)$. For any \bi-lineage polynomial $\poly(\vct{X})$ we have
  % The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
\begin{equation*}
\expct_{\vct{X}}\pbox{\poly(\vct{X})}  = \rpoly(\vct{p}).
\end{equation*}
\end{Lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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 a \bi-lineage polynomial and evaluate the resulting expression in $\mathbb{R}$, then the result is the expectation of the polynomial.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof for Lemma ~\ref{lem:exp-poly-rpoly}]
%Using the fact above, we need to compute \[\sum_{(\wbit_1,\ldots, \wbit_\numvar) \in \{0, 1\}}\rpoly(\wbit_1,\ldots, \wbit_\numvar)\].  We therefore argue that
%\[\sum_{(\wbit_1,\ldots, \wbit_\numvar) \in \{0, 1\}}\rpoly(\wbit_1,\ldots, \wbit_\numvar) = 2^\numvar \cdot \rpoly(\frac{1}{2},\ldots, \frac{1}{2}).\]

Let $\poly$ be the generalized polynomial, i.e., the polynomial of $\numvar$ variables with highest degree $= B$: %, in which every possible monomial permutation appears,
\[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i}\].


Then, assigning $\vct{w}$ to $\vct{X}$, for expectation we have
\begin{align}
\expct_{\vct{w}}\pbox{\poly(\vct{w})} &= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \expct_{\vct{w}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar w_i^{d_i}}\label{p1-s1}\\
&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{w}}\pbox{w_i^{d_i}}\label{p1-s2}\\
&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{w}}\pbox{w_i}\label{p1-s3}\\
&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \prob_i\label{p1-s4}\\
&= \rpoly(\prob_1,\ldots, \prob_\numvar)\label{p1-s5}
\end{align}

In steps \cref{p1-s1} and \cref{p1-s2}, by linearity of expectation (recall the variables are independent), the expecation can be pushed all the way inside of the product.  In \cref{p1-s3}, note that $w_i \in \{0, 1\}$ which further implies that for any exponent $e \geq 1$, $w_i^e = w_i$.  Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.

%\OK{
%	You don't need to tie this to TI-DBs if you define the variables ($X_i$) to be independent.
%	Annotations
%	Boolean expressions over uncorrelated boolean variables are sufficient to model TI-, BI-, and
%	PC-Tables.  This should still hold for arithmetic over the naturals.
%}


Finally, observe \cref{p1-s5} by construction in \cref{lem:pre-poly-rpoly}, that $\rpoly(\prob_1,\ldots, \prob_\numvar)$ is exactly the product of probabilities of each variable in each monomial across the entire sum.

\qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Corollary}\label{cor:expct-sop}
If $\poly$ is given as a sum of monomials, the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $O(|\poly|)$, where $|\poly|$ denotes the total number of multiplication/addition operators in $\poly$.
\end{Corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof For Corollary ~\ref{cor:expct-sop}]
Note that \cref{lem:exp-poly-rpoly} shows that $\expct\pbox{\poly} =$ $\rpoly(\prob_1,\ldots, \prob_\numvar)$.  Therefore, if $\poly$ is already in sum of products form, one only needs to compute $\poly(\prob_1,\ldots, \prob_\numvar)$ ignoring exponent terms (note that such a polynomial is $\rpoly(\prob_1,\ldots, \prob_\numvar)$), which indeed has $O(|\poly|)$ compututations.\qed
\end{proof}
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