diff --git a/poly-form.tex b/poly-form.tex
index ba9f5e0..1618d43 100644
--- a/poly-form.tex
+++ b/poly-form.tex
@@ -7,6 +7,7 @@ Let $\vect_1,\ldots, \vect_\numTup$ be vectors annotating $\numTup$ tuples in a
 \end{cases}
 \end{equation}
 Here we define vector indexing by the $\numTup$-bit binary tuple $\wVec = (\wbit_1,\ldots,\wbit_\numTup)$ such that the possible world $\wVec$ is identified by its bit vector binary value.
+\AR{I do not see why we need to define $\vect$-- everything can be defined without bringing the $\vect$ in. I would recommend that in Section 1 you define things without going into $\vect$-- i.e. state the DB queries and TIDBs and directly define the query polynomial.}
 
 
 %---We have chosen to ignore the vector formulation
@@ -19,9 +20,19 @@ Define $\poly(X_1,\ldots, X_\numTup)$ as a polynomial whose variables represent
 
 \[\expct_{\wVec}\pbox{\poly(\wVec)} = \sum\limits_{\wVec \in \{0, 1\}^\numTup} \poly(\wVec)\prod_{\substack{i \in [\numTup]\\ s.t. \wElem_i = 1}}\prob_i \prod_{\substack{i \in [\numTup]\\s.t. w_i = 0}}\left(1 - \prob_i\right).\]
 
-Further, define $\rpoly(X_1,\ldots, X_\numTup)$ as the reduced version of $\poly(X_1,\ldots, X_\numTup)$, of the form
-\[\rpoly(\wbit_1,\ldots, \wbit_\numTup) = \poly(\wbit_1,\ldots, \wbit_\numTup) \mod \wbit_1^2-\wbit\cdots\mod \wbit_\numTup^2 - \wbit_\numTup.\]  Intuitively, $\rpoly(\wVec)$ is the expanded sum of products form of $\poly(\wVec)$ such that if any $\wbit_j$ term  has an exponent $e > 1$, it is reduced to $1$, i.e. $\wbit_j^e\mapsto \wbit_j$ for any $e > 1$.  The usefulness of this reduction will be seen shortly.
+\AR{The above should go into Section 1 (without using $\vect$  of course). And as I mentioned in my comment in Sec 1, you need to figure out a notation for the queries. Check with Oliver on what is standard notation in the PDB literature (unless you know the standard  notation in which case eno need to ask Oliver :-).
 
+Also it might be worthwhile to define a notation for the probability that the world is the specific $\wVec$-- then you can define the expectation for PDB models  other than TIDBs.}
+
+Further, define $\rpoly(X_1,\ldots, X_\numTup)$ as the reduced version of $\poly(X_1,\ldots, X_\numTup)$, of the form
+\[\rpoly(\wbit_1,\ldots, \wbit_\numTup) = \poly(\wbit_1,\ldots, \wbit_\numTup) \mod \wbit_1^2-\wbit\cdots\mod \wbit_\numTup^2 - \wbit_\numTup.\]  
+\AR{the $w_i$'s should be $X_i$'s. A general comment: to make things clearer, always use $X_i$'s to denote the variabls and $w_i$'s to denote the values that we substitute the variables with.}
+Intuitively, $\rpoly(\wVec)$ is the expanded sum of products form of $\poly(\wVec)$ such that if any $\wbit_j$ term  has an exponent $e > 1$, it is reduced to $1$, i.e. $\wbit_j^e\mapsto \wbit_j$ for any $e > 1$.  The usefulness of this reduction will be seen shortly.
+\AR{The intuition above should be given for the variable setting: i.e. using $X_i$ instead of $w_i$.}
+
+\AR{You should first state a lemma that show what $\rpoly$ looks like given  $\poly(X_1,\ldots, X_\numTup) = \sum_{\vct{d} \in \{0,\ldots, D\}^\numTup}q_{\vct{d}}\cdot \prod_{i = 1\text{ s.t. }d_i \geq 1}^\numTup X_i^{d_i}$.}
+
+\AR{The statement below should be typeset as a proposition.}
 First, note the following fact:
 \[\text{For all } (\wbit_1,\ldots, \wbit_\numTup) \in \{0, 1\}^\numTup, \poly(\wbit_1,\ldots, \wbit_\numTup) = \rpoly(\wbit_1,\ldots, \wbit_\numTup).\]
 
@@ -29,7 +40,7 @@ First, note the following fact:
 For all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = 1$.\qed
 \end{proof}
 
-Assuming each tuple has a probability $\prob = \frac{1}{2}$, we note that
+\AR{The statement below should be a lemma.}
 \begin{Property}\label{prop:l1-rpoly-numTup}
 The expectation of a possible world in $\poly$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numTup)$.
 \begin{equation*}