comments section 1&2

intro.tex

@@ -97,7 +97,7 @@ In other words, our approximation algorithm can estimate expected multiplicities
 \begin{Example}\label{ex:intro}
 Consider the Tuple Independent ($\ti$) Set-PDB\footnote{Our work also handles Block Independent Disjoint Databases ($\bi$)~\cite{DBLP:conf/sigmod/BoulosDMMRS05,DBLP:series/synthesis/2011Suciu}, we return to this model later.} given in \Cref{fig:intro-ex} with two input relations $R$ and $E$.
 Each input tuple is assigned an annotation (attribute $\Phi$): an independent random boolean variable ($W_i$) or the constant $\top$.
-Each assignment of values to variables ($\{\;W_a,W_b,W_c\;\}\mapsto \{\;\top,\bot\;\}$) identifies one \emph{possible world}, a deterministic database instance that contains exactly the tuples annotated by the constant $\top$ or by a variable assigned to $\top$.
+Each assignment of values to variables ($\{\;W_a,W_b,W_c\;\}\mapsto \{\;\top,\bot\;\}$) \SF{Do we need to state the meaning of $\top$ and $\bot$? Also do we want to add bag annotation to Figure 1 too since we are discussing both sets and bags later?} identifies one \emph{possible world}, a deterministic database instance that contains exactly the tuples annotated by the constant $\top$ or by a variable assigned to $\top$.
 The probability of this world is the joint probability of the corresponding assignments.  
 For example, let $P[W_a] = P[W_b] = P[W_c] = p$ and consider the possible world where $R = \{\;\tuple{a}, \tuple{b}\;\}$.  
 The corresponding variable assignment is $\{\;W_a \mapsto \top, W_b \mapsto \top, W_c \mapsto \bot\;\}$, and the probability of this world is $P[W_a]\cdot P[W_b] \cdot P[\neg W_c] = p^2-p^3$ 
@@ -243,6 +243,7 @@ With $\poly^2$ as an example, we have:
 &+ 2W_aW_bW_c\\
 =&\; W_aW_b + W_bW_c + W_cW_a + 6W_aW_bW_c
 \end{align*}
+\SF{Should this be like $\tilde{\poly^2}$ to avoid ambiguous?}
 Observe that the reduced polynomial is a closed form formula for the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(P\pbox{W_a=1}, P\pbox{W_b=1}, P\pbox{W_c=1})$).
 Also note that our initial example polynomial $\poly$ is already in reduced form.
 

poly-form.tex

@@ -50,9 +50,10 @@ The degree of the running example polynomial is $2$. In this paper we consider o
 We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (\emph{\ti-lineage polynomial}), if
 \AH{Why is it required for the tuple to be n-ary?  I think this slightly confuses me since we have n tuples.} 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 $\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\prob(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\prob$ and the blocks induce a the probability distribution $\pd$ of $\pxdb$.
+Recall that in a \bi $\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\prob(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\prob$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
 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 $[\abs{\block_i}]$.
+\SF{Where is $\block_{i, j}$ used? Is it $X_{\block_{1, 1}}$ or $X_{\block_1, 1}$ ?}
 % and the probability distribution of $\pxdb$ 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 $\pxdb$.