Fixed Lemma 2.8 proof.

app_notation-background.tex

@@ -160,16 +160,18 @@ Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion
 \begin{proof}
 Let $\poly$ be the generalized polynomial, i.e., the polynomial of $\numvar$ variables with highest degree $= B$: %, in which every possible monomial permutation appears,
 \[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i}.\]
-Then, %denoting the corresponding exponent vector $\vct{d}$ for a world $\vct{\wElem}$ over the set of valid worlds $\valworlds$ as $\vct{d} \in \valworlds$, 
-in expectation we have
+Let the boolean function $\isInd{\cdot}$ take $\vct{d}$ as input and return true if there does not exist any dependent variables in $\vct{d}$, i.e., $\not\exists ~\block, i\neq j\suchthat d_{\block, i}, d_{\block, j} \geq 1$.\footnote{This \abbrBIDB notation is used and discussed in \cref{subsec:tidbs-and-bidbs}}.
+Then in expectation we have
 \begin{align}
-\expct_{\vct{W}}\pbox{\poly(\vct{W})} &= \sum_{\vct{d} \in \{0,\ldots,B\}^\numvar}c_{\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}c_{\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}c_{\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}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \prob_i\label{p1-s4}\\
+\expct_{\vct{\randWorld}}\pbox{\poly(\vct{\randWorld})} &= \expct_{\vct{\randWorld}}\pbox{\sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i} + \sum_{\substack{\vct{d} \in \{0,\ldots, B\}^\numvar\\\wedge ~\neg\isInd{\vct{d}}}} c_{\vct{d}}\cdot\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar\randWorld_i^{d_i}}\label{p1-s1a}\\
+&= \sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \expct_{\vct{\randWorld}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i}} + \sum_{\substack{\vct{d} \in \{0,\ldots, B\}^\numvar\\\wedge ~\neg\isInd{\vct{d}}}} c_{\vct{d}}\cdot\expct_{\vct{\randWorld}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar\randWorld_i^{d_i}}\label{p1-s1b}\\
+&= \sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\~\wedge\isInd{\vct{d}}}}c_{\vct{d}}\cdot \expct_{\vct{\randWorld}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i}}\label{p1-s1c}\\
+&= \sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{\randWorld}}\pbox{\randWorld_i^{d_i}}\label{p1-s2}\\
+&= \sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{\randWorld}}\pbox{\randWorld_i}\label{p1-s3}\\
+&= \sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\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}
-\Cref{p1-s1} is the result of the following facts.  First, the only worlds contributing to the expectation are the valid worlds, i.e. those worlds each of which produce a polynomial whose monomials are all made up of independent variables.  Second, linearity of expectation combined with the fact that any non-random variable can be pulled out of the expectation allow for the expectation to be pushed through the sum and coefficient.  \Cref{p1-s2} is obtained by the independence property of \abbrBIDB\xplural, where any valid possible world is made up of independent tuples, and this allows for the expectation to be pushed through 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.
+\Cref{p1-s1a} is the result of substituting in the definition of $\poly$ given above.  Then we arrive at \cref{p1-s1b} by linearity of expectation.  Next, \cref{p1-s1c} is the result of the independence constraint of \abbrBIDB\xplural, specifically that no monomial can be composed of dependent variables, i.e., variables from the same block $\block$.\Cref{p1-s2} is obtained by the fact that all variables in each monomial are independent, which allows for the expectation to be pushed through the product.  In \cref{p1-s3}, note that $\randWorld_i \in \{0, 1\}$ which further implies that for any exponent $e \geq 1$, $\randWorld_i^e = \randWorld_i$.  Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.
 
 Finally, observe \Cref{p1-s5}, where 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 and its corresponding coefficient, across the entire sum.
 \qed

macros.tex

@@ -73,6 +73,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newcommand{\domain}{\func{Dom}}
 \newcommand{\func}[1]{\textsc{#1}}
+\newcommand{\isInd}[1]{\func{isInd}\inparen{#1}}
 \newcommand{\polyf}{\func{poly}}
 \newcommand{\evalmp}{\func{eval}}
 \newcommand{\degree}{\func{deg}}