More rebuttals; had to add some text to S1 and S2.

intro-rewrite-070921.tex

@@ -15,7 +15,7 @@ The natural generalization of the problem of computing marginal probabilities of
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Problem}[Expected Multiplicity]\label{prob:bag-pdb-query-eval}
-Given an \raPlus query\footnote{The class of positive relational algebra (\raPlus) queries consists of all queries that can be composed of the positive (monotonic) relational algebra operators: selection, projection, join, and union (SPJU).} $\query$, \abbrBPDB $\pdb$, and result tuple $\tup$, compute the expected
+Given an \raPlus query\footnote{The class of positive relational algebra (\raPlus) queries consists of all queries that can be composed of the positive (monotonic) relational algebra operators: selection, projection, join, and union (SPJU).\label{footnote:ra-def}} $\query$, \abbrBPDB $\pdb$, and result tuple $\tup$, compute the expected
 multiplicity ($\expct_{\db\sim\pd}\pbox{\query\inparen{\db}\inparen{\tup}}$)
 of tuple $\tup$.
 \end{Problem}

ra-to-poly.tex

@@ -39,7 +39,7 @@ Recall \Cref{fig:nxDBSemantics} which depicts the semantics for constructing a l
 
 \begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
 Given a \abbrBPDB $\pdb = (\idb,\pd)$ and lineage polynomial $\apolyqdt$ for aribitrary output tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, 
-we have:
+we have (denoting $\randDB$ as the random variable over $\idb$):
   $ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
 \end{Proposition}
 \noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.\footnote{Although \Cref{prop:expection-of-polynom} follows, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}

rebuttal.tex

@@ -133,7 +133,43 @@ We have added a reference. Please see \Cref{lem:val-ub}.
 \AH{Needs to be addressed.}
 
 \RCOMMENT{In section 5, it seems you are arguing that we can compute lineages as arithmetic circuits at the same time as we would be running an ordinary query evaluation plan. How is that different from using the relations in Figure 2 for computing the lineage?}
-There is not a major difference between the two.  We explicitly focus on circuits since our approximation results rely on them.  We have taken pains to be clear that our hardness results do not rely on them, while our approximation results do.  This can be seen e.g. in the progressional sequence of problem statements in the revised introduction (\Cref{sec:intro}).
+There is not a major difference between the two.  This observation has persuaded us to eliminate $\semNX$-DB query evaluation and have only an algorithm for lineage.  
+
+\RCOMMENT{l.679 where do you use $max(D_i)$ later in the proof?}
+\AH{Needs to be fixed.}
+
+\RCOMMENT{l.688 That sentence is hard to parse, consider reformulating it}
+\AH{Needs to be reformulated.} 
+
+\RCOMMENT{it seems you are defining N[X]-PDB at two places in the appendix: once near l.632, and another time near l.652}
+\AH{Needs to be addressed.}
+
+
+\subsection{Reviewer 2}
+\RCOMMENT{First, the paper should state rigorously the problem definition.  There are three well-known definitions in database theory: data complexity, combined complexity, and parameterized complexity.  If I understand correctly, Theorem 3.4 refers to the parameterized complexity, Theorem 3.6 refers to the data complexity (of a fixed query), while the positive results in Sec. 4 (e.g. Th. 4.8) introduce yet another notion of complexity, which requires discussion.}
+We have addressed the concerns in rewriting the entirety of \Cref{sec:intro}, explicitly mentioning complexity metrics considered, while forming a series of problem statements that intuitively describe the exact problem we are considering, and the complexity metrics considered.  We have also adjusted the phrasing of the said theorems and definitions to eliminate the ambiguity described.
+
+\RCOMMENT{The problem definition is supposed to be in Definition 2.14, but this definition is sloppy.  It states that the input to the problem is a circuit C: but then, what is the role of the PDB and the query Q?  Currently Definition 2.14 reads as follows: "Given a circuit C defining some polynomial Q(X), compute E[Q(W)]", and, thus, the PDB and the query play no role at all.  All results in Section 4 seem to assume this simplified version of Definition 2.14.  On the other hand, if one interprets the definition in the traditional framework of data complexity (Q is fixed, the inputs are D and C) then the problem is solvable in PTIME (and there is no need for C), since E[Q(W)] is the sum of expectations of the monomials in Q (this is mentioned in Example 1.2).}
+We have rephrased \Cref{def:the-expected-multipl} to qualify data complexity.  The paper (especially in \Cref{sec:intro}) builds up the fact that we aren't stopping at polynomial time, but exploring parameterized complexity and fine grained analysis (as the reviewer aptly noted in the first comment).
+
+\RCOMMENT{Second, Definition 2.6 of Reduced BIDB polynomials is simply wrong.  It uses "mod" of two multivariate polynomials, but "mod" doesn't exists for multivariate polynomials...Either state Definition 2.6 directly, in an ad-hoc manner (which seems doable), or do a more thorough job grounding it in the ring of multivariate polynomials and its ideals.
+}
+We have implemented the reviewer's ad-hoc suggestion in light of Reviewer 1's similar suggestions.
+
+\RCOMMENT{the paper uses three notations (UCQ, RA+, SPJU) for the same thing, and never defines formally any of them.}
+We have chosen $\raPlus$ for consistent use throughout the paper.  We have included \Cref{footnote:ra-def} on \Cpageref{footnote:ra-def} for an explicity definition of $\raPlus$ queries.
+
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