Minor changes.

macros.tex

@@ -100,6 +100,8 @@
 \newcommand{\pxdb}{\pdb_{\semNX}}%<---changed from the origninal \mathbf{\db}
 \newcommand{\nxdb}{D(\vct{X})}%\mathbb{N}[\vct{X}] db--Are we currently using this?
 
+\newcommand{\valworlds}{\eta}
+
 %BIDB
 \newcommand{\block}{b}
 \newcommand{\bivar}{x_{\block, i}}
@@ -181,6 +183,8 @@
 \newcommand{\polyX}{\poly\inparen{\pVar}}%<---let's see if this proves handy
 \newcommand{\rpoly}{\widetilde{\poly}}%r for reduced as in reduced 'Q'
 \newcommand{\rpolyX}{\rpoly\inparen{\pVar}}%<---if this isn't something we use much, we can get rid of it
+\newcommand{\biDisProd}{\mathcal{B}}%bidb disjoint tuple products (def 2.5)
+\newcommand{\rExp}{\mathcal{T}}%the set of variables to reduce all exponents to 1 via modulus operation; I think \mathcal T collides with the notation used for the set of tuples in D
 \newcommand{\polyForTuple}{\poly_{\tup}}%do we use this?
 %Do we use this?
 \newcommand{\out}{output}%output aggregation over the output vector

poly-form.tex

@@ -111,10 +111,10 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Valid Worlds]
-For probability distribution $\probDist$, % and its corresponding probability mass function $\probOf$,
-the set of valid worlds $\eta$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
+For probability distribution $\pd$, % and its corresponding probability mass function $\probOf$,
+the set of valid worlds $\valworlds$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
 \[
-\eta = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
+\valworlds = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
 \]
 \end{Definition}
 
@@ -133,10 +133,10 @@ the set of valid worlds $\eta$ consists of all the worlds with probability value
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Lemma}\label{lem:exp-poly-rpoly}
-Let $\pxdb$ be a \bi over variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ and with probability distribution $\probDist$ produced by the tuple probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$ over all $\vct{w}$ in $\eta$. For any \bi-lineage polynomial $\poly(\vct{X})$ based on $\pxdb$ and query $\query$ we have:
+Let $\pxdb$ be a \bi over variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ and with probability distribution $\pd$ induced by the tuple probability vector $\probAllTup = (\prob_1, \ldots, \prob_\numvar)$ over all $\vct{w}$ in $\valworlds$. For any \bi-lineage polynomial $\poly(\vct{X})$ based on $\pxdb$ and query $\query$ we have:
   % The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
 \begin{equation*}
-\expct_{\vct{W}\sim \probDist}\pbox{\poly(\vct{W})}  = \rpoly(\probAllTup).
+\expct_{\vct{W}\sim \pd}\pbox{\poly(\vct{W})}  = \rpoly(\probAllTup).
 \end{equation*}
 \end{Lemma}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

prob-def.tex

@@ -11,9 +11,9 @@ We represent query polynomials via {\em arithmetic circuits}~\cite{arith-complex
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Circuit]\label{def:circuit}
-A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source nodes (in degree of $0$) consist of elements in either $\domR$ or $\vct{X}$.  The internal nodes and (the single) sink node of $\circuit$ (corresponding to the result tuple $t$) have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
+A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domR$ or $\vct{X}$.  The internal gates and (the single) sink gate of $\circuit$ (corresponding to the result tuple $t$) have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
 %
-Each node in a circuit $\circuit$ has the following members: \type, \val, \vpartial, \vari{input}, \degval and \vari{Lweight}, \vari{Rweight}, where \type is the type of value stored in the node (one of $\{\circplus, \circmult, \var, \tnum\}$, \val is the value stored (a constant or variable), and \vari{input} is the list of the nodes inputs. We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input or the sink of circuit $\circuit$.
+Each node in a circuit $\circuit$ has the following members: \type, \val, \vpartial, \vari{input}, \degval and \vari{Lweight}, \vari{Rweight}, where \type is the type of value stored in the gatee (one of $\{\circplus, \circmult, \var, \tnum\}$, \val is the value stored (a constant or variable), and \vari{input} is the list of the gate's inputs. We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input of the sink of circuit $\circuit$.
 %The member \degval holds the degree of \circuit.
 When the underlying DAG is a tree (with edges pointing towards the root), we will refer to the structure as an expression tree \etree.  Note that in such a case, the root of \etree is analogous to the sink of \circuit.
 \end{Definition}