Changes to S.2.

poly-form.tex

@@ -26,13 +26,14 @@ When it is unclear, we use $\smbOf{\poly}$ to denote the \abbrSMB form of a poly
 The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such that $c_{(d_1,\dots,d_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
+\AH{The example needs to change to avoid ambiguity between different definitions of \emph{degree}.  Our definition is the one we want, since we two different tuples join without any tuples joining on themselves, and the degree would be $2$, rather than $1$, for example.}
 The degree of the polynomial $X^2+2XY+Y^2$ is $2$.
 Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins in any clause of the UCQ query that created it.
 In this paper we consider only finite degree polynomials.
 We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a \AH{Which formalism?  UCQ?}$\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\poly\inparen{\vct{X}} = \query(\pxdb)(\tup)$. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\AH{There was reviewer disgruntlement with how we discussed modding and all in the next few definitions.  Need to revisit these comments and adjust accordingly.}
 \begin{Definition}[Modding with a set]\label{def:mod-set}
 Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
 \end{Definition}
@@ -158,6 +159,7 @@ to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that w
 \begin{Corollary}\label{cor:expct-sop}
 If $\poly$ is a \bi-lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $\bigO{\size\inparen{\poly}}$, where $\size\inparen{\poly}$ (\Cref{def:size}) is proportional to the total number of multiplication/addition operators in $\poly$.
 \end{Corollary}
+\AH{Maybe state in terms of $\abs{\circuit}$.}
 %\AH{What if $\poly$ is not in \abbrSMB form?}
 
 

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 gates (in degree of $0$) consist of elements in either $\domN$ 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.
+A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$.  The internal gates 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 gate (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 $\circuit$.
+Each internal node in a circuit $\circuit$ has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the type of value stored in the gate (one of $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} is the list of the gate's inputs. The source gates have an additional member \val, which holds the value stored (constant or variable).  We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input of a 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}
@@ -22,7 +22,7 @@ When the underlying DAG is a tree (with edges pointing towards the root), we wil
 
 As stated in \Cref{def:circuit}, every internal node has at most two in-edges, is labeled as an addition or a multiplication node, and has no limit on its outdegree.
 Note that if we limit the outdegree to one, then we get expression trees.
-We ignore the fields \vari{partial}, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{We omit degree here too, which {\emph I think} is used only in appendix proofs.}
+We ignore the fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{We omit degree here too, which {\emph I think} is used only in appendix proofs.}
 
 
 \begin{Example}

ra-to-poly.tex

@@ -5,7 +5,7 @@
 
 \subsection{Probabilistic Databases}
 
-We focus primarily on set-\abbrPDB inputs in this section, but as noted in \cref{sec:intro-rewrite-070921}, this is not limiting.  
+The setting used in this section is primarily that of a bag-\abbrPDB query with set-\abbrPDB inputs.  Recall, as noted in \cref{sec:intro-rewrite-070921}, this is not limiting.  
 
 An \textit{incomplete database} $\idb$ is a set of deterministic databases $\db$ called possible worlds.
 Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\pdb$ is a pair $(\idb, \pd)$ where $\idb$ is an incomplete database and $\pd$ is a probability distribution over $\idb$. Queries over probabilistic databases are evaluated using the so-called possible world semantics. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\idb$ is the set of query answers produced by evaluating $\query$ over each possible world: $\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}$.
@@ -47,7 +47,7 @@ A \emph{\ti} is a \bi where each block contains exactly one tuple.
 \Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
 %
 In a \bi (and by extension a \ti) $\pxdb$, tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
-  Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function).  For $t_j \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
+  Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function).  For $t_{i, j} \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
 }
 Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ 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