Cleaning Appendix A up to Lem 1.4.

app_notation-background.tex

@@ -63,7 +63,7 @@ Closure under $\raPlus$ queries follows from this and from \cite{DBLP:conf/pods/
 \subsubsection{\tis and \bis in the \abbrNXPDB model}\label{subsec:supp-mat-ti-bi-def}
 
 Two important subclasses of \abbrNXPDB\xplural that are of interest to us are the bag versions of tuple-independent databases (\tis) and block-independent databases (\bis). Under set semantics, a \ti is a deterministic database $\db$ where each tuple $\tup$ is assigned a probability $\prob_\tup$. The set of possible worlds represented by a \ti $\db$ is all subsets of $\db$. The probability of each world is the product of the probabilities of all tuples that exist with one minus the probability of all tuples of $\db$ that are not part of this world, i.e., tuples are treated  as independent  random events. In a \bi, we also  assign each tuple a  probability,  but  additionally partition  $\db$ into blocks. The possible worlds of a \bi $\db$ are all subsets  of $\db$ that contain at most one tuple  from each block.  Note then that the tuples sharing the same block are disjoint, and the sum of the probabilitites of all the tuples in the same block $\block$ is at most $1$.
-The probability of such a world is the product of the probabilities of all tuples present in the world. 
+The probability of such a world is the product of the probabilities of all tuples present in the world and the product of the probabilities that no tuple is present in each block $\block$ for which no tuple exists in that world. 
 For bag \tis and \bis, we define the probability of a tuple to  be the probability that the tuple exists with multiplicity at least $1$.
 
 In this work, we define \tis and \bis as subclasses of \abbrNXPDB\xplural defined over variables $\vct{X}$ (\Cref{def:semnx-pdbs}) where $\vct{X}$ can be partitioned into blocks that satisfy the conditions of a \ti or \bi (stated formally in \Cref{subsec:tidbs-and-bidbs}).
@@ -85,7 +85,7 @@ Recall that tuple blocks in a TIDB always have size 1, so the outer summation of
 \subsection{Proof of~\Cref{prop:expection-of-polynom}}
 \label{subsec:expectation-of-polynom-proof}
 \begin{proof}
-We need to prove for $\semN$-PDB $\pdb = (\idb,\pd)$ and \abbrNXPDB $\pxdb = (\db_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$ that $\expct_{\randDB\sim \pd}[\query(\db)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\nxpolyqdt(\vct{W})}$
+We need to prove for $\semN$-PDB $\pdb = (\idb,\pd)$ and \abbrNXPDB $\pxdb = (\db_{\semNX},\pd')$ where $\rmod(\pxdb) = \pdb$ that $\expct_{\randDB\sim \pd}[\query(\db)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\nxpolyqdt(\vct{W})}$
 By expanding $\nxpolyqdt$ and the expectation we have:
 \begin{align*}
 \expct_{\vct{W} \sim \pd'}\pbox{\poly(\vct{W})}
@@ -106,18 +106,19 @@ By expanding $\nxpolyqdt$ and the expectation we have:
 \noindent Note the following fact:
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{W}$ such that $\probOf[\vct{W} = \vct{W}] > 0$, it holds that
+\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{W}$ such that $\probOf\pbox{\vct{W}} > 0$, 
+it holds that
 $%  \[
     \poly(\vct{W}) = \rpoly(\vct{W}).
 $%    \]
 \end{Proposition}
 
 \begin{proof}
-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 $e \geq 1$, $b^e = b$.
-Finally, note that there are exactly three cases where the expectation of a monomial term $\expct\left[c_{\vct{d}}\prod_{i = 1\; s.t.\; \vct{d}_i \geq 1}^\numvar X_i\right]$ is zero:
-(i) when $c_{\vct{d}} = 0$,
-(ii) when $p_i = 0$ for some $i$ where $\vct{d}_i \geq 1$, and
-(iii) when $X_i$ and $X_j$ are in the same block for some $i,j$ where $\vct{d}_i, \vct{d}_j \geq 1$.
+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 $e \geq 1$, $b^e = b$.  By definition (see~\Cref{def:reduced-poly-one-bidb}), $\rpoly\inparen{\vct{X}}$ is the \abbrSMB expansion of $\poly\inparen{\vct{X}}$ followed by reducing every exponent $e > 1$ to $1$ and eliminating all cross terms for the \abbrBIDB case.  Note that it must be that no cross terms exist in $\poly\inparen{\vct{X}}$, since by the proposition statement, $\probOf\pbox{\vct{W}} > 0$.  Thus, since all monomials are indeed the same, it follows that $\poly\inparen{\vct{W}} = \rpoly\inparen{\vct{W}}$.
+%Finally, note that there are exactly three cases where the expectation of a monomial term $\expct\left[c_{\vct{d}}\prod_{i = 1\; s.t.\; \vct{d}_i \geq 1}^\numvar X_i\right]$ is zero:
+%(i) when $c_{\vct{d}} = 0$,
+%(ii) when $p_i = 0$ for some $i$ where $\vct{d}_i \geq 1$, and
+%(iii) when $X_i$ and $X_j$ are in the same block for some $i,j$ where $\vct{d}_i, \vct{d}_j \geq 1$.
 \qed
 \end{proof}