poly

poly-form.tex

@@ -4,43 +4,59 @@
 \subsection{Polynomial Formulation and Equivalences}
 
 Since we have shown that computing the expected multiplicity of a 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.
-Before proceeding, note that the following is assuming \bis (which subsume \tis as a special case). Thus, variables are independent of each other and each variable $X$ is associated with a probability $\vct{p}(X)$.
-
-Let us use the expression $(x + y)^2$ for a running example in the following definitions.
+Before proceeding, note that the following is assuming \bis (which subsume \tis as a special case). Thus, variables are independent of each other and each variable $X$ is associated with a probability $\vct{p}(X) = \pd[X = 1]$.
+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 fixed set of variables, each raised to a non-negative integer power.
+A monomial is a product of a set of variables, each raised to a non-negative integer power.
 \end{Definition}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-For the term $2xy$, by ~\cref{def:monomial} the monomial is $xy$.
+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 polynomial is in standard monomial basis when it is fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
+  A polynomial is in standard monomial basis 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 $i \neq j$.
+%  fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
 \end{Definition}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The standard monomial basis 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.
 
 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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 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}[Degree]\label{def:degree}
 The degree of polynomial $\poly(\vct{X})$ is the maximum sum of the exponents of a monomial, over all monomials when $\poly(\vct{X})$ is in SOP form.
 \end{Definition}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The degree of the running example is $2$.  In this paper we consider only finite degree polynomials.
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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 when $\poly(x, y) = (x + y)(x + y)$.  Then the expanded derivation for $\rpoly(x, y)$ is
 \begin{align*}
@@ -49,12 +65,14 @@ Consider when $\poly(x, y) = (x + y)(x + y)$.  Then the expanded derivation for
 = ~& x + 2xy + y
 \end{align*}
 \end{Example}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 Intuitively, $\rpoly(\textbf{X})$ is the SOP 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 $e > 1$.
 Alternatively, one can gain intuition for $\rpoly$ by thinking of $\rpoly$ as the resulting SOP of $\poly(\vct{X})$ with an idemptent product operator.
 
 When considering $\bi$ input, it becomes necessary to redefine $\rpoly(\vct{X})$.
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[$\rpoly$ $\bi$ Redefinition]
 A polynomial $\poly(\vct{X})$ over a $\bi$ instance is reduced to $\rpoly(\vct{X})$ with the following criteria.  First, all exponents $e > 1$ are reduced to $e = 1$.  Second, all monomials sharing the same $\block$ are dropped.  Formally this is expressed as
 
@@ -63,36 +81,44 @@ A polynomial $\poly(\vct{X})$ over a $\bi$ instance is reduced to $\rpoly(\vct{X
 \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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The usefulness of this reduction will be seen 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:qtilde}.
+Follows by the construction of $\rpoly$ in \cref{def:qtilde}. \qed
 \end{proof}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\qed
 
 Note the following fact:
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Proposition}\label{proposition:q-qtilde}
 \[\text{For all } (X_1,\ldots, X_\numvar) \in \{0, 1\}^\numvar, \poly(X_1,\ldots, X_\numvar) = \rpoly(X_1,\ldots, X_\numvar).\]
 \end{Proposition}
-
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{proof}[Proof for Proposition ~\ref{proposition:q-qtilde}]
-Note that any $\poly$ in factorized form is equivalent to its sum of product expansion.  For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$.
+Note that any $\poly$ in factorized form is equivalent to its sum of product 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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\qed
 
 Define all variables $X_i$ in $\poly$ to be independent.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Lemma}\label{lem:exp-poly-rpoly}
 The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
 \begin{equation*}
 \expct_{\vct{w}}\pbox{\poly(\vct{w})}  = \rpoly(\prob_1,\ldots, \prob_\numvar).
 \end{equation*}
 \end{Lemma}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 Note that in the preceding lemma, we have assigned $\vct{p}$ (introduced in ~\cref{subsec:def-data}) to the variables $\vct{X}$.
 
@@ -127,11 +153,19 @@ Finally, observe \cref{p1-s5} by construction in \cref{lem:pre-poly-rpoly}, that
 
 \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.
 \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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