Rebuttal-oriented fixes

app_notation-background.tex

@@ -110,7 +110,9 @@ By expanding $\nxpolyqdt$ and the expectation we have:
 & = \sum_{\vct{w} \in \{0,1\}^n}\probOf(\vct{w}) \cdot Q(\db_{\semNX})(t)(\vct{w})\\
 \intertext{From $\rmod(\pxdb) = \pdb$, we have that the range of $\assign_{\vct{w}(\pxdb)}$ is $\idb$, so}
 & = \sum_{\db \in \idb}\;\;\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w}) \cdot Q(\db_{\semNX})(t)(\vct{w})\\
-\intertext{In the inner sum, $\assign_{\vct{w}}(\pxdb) = \db$, so by distributivity of $+$ over $\times$}
+\intertext{The inner sum is only over $\vct{w}$ where $\assign_{\vct{w}}(\pxdb) = \db$ (i.e., $Q(\db_{\semNX})(t)(\vct{w}) = \query(\db)(t))$}
+& = \sum_{\db \in \idb}\;\;\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w}) \cdot \query(\db)(t)\\
+\intertext{By distributivity of $+$ over $\times$}
 & = \sum_{\db \in \idb}\query(\db)(t)\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w})\\
 \intertext{From the definition of $\pd$ in \cref{def:semnx-pdbs}, given $\rmod(\pxdb) = \pdb$, we get}
 & = \sum_{\db \in \idb}\query(\db)(t) \cdot \probOf(D) \quad = \expct_{\randDB \sim \pd}[\query(\db)(t)]

rebuttal.tex

@@ -117,7 +117,10 @@ The new text precisely defines TIDBs (\Cref{sec:intro}), and the BIDB generaliza
 The reviewer is correct and we have updated our appendix text accordingly.
 
 \RCOMMENT{it is not clear to me how you can go from l.733 to l.736, which is sad because this is actually the whole point of this proof.  If I understand correctly, in l.733, Q(D)(t) is the polynomial annotation of t when you use the semantics of Figure 2 with the semiring K being N[X], so I don't see how you go from this to l.736}
-\AH{Needs to be verified.  I have looked at this previously, and the proof iirc.}
+
+This result follows from the inner sum looping only over $\vct{w}$ s.t. $\assign_{\vct{w}}(\pxdb) = \db$.  As a consequence of this constraint, we have $Q(\db_{\semNX})(t)(\vct{w}) = \query(\db)(t)$.  The latter term is independent of the summation, and so can be pulled out by distributivity of addition over multiplication.
+
+We agree with the reviewer that this could be presented more clearly, and have now split the distributivity argument into a separate step.
 
 \RCOMMENT{l.209-227: so you define what is a polynomial and what is the degree of a polynomial (things that everyone knows), but you don't bother explaining what "taking the mod of Q(X) over all polynomials in S" means? This is a bit weird.}
 Based on this and other reviewer comments, we removed the formal definition of $\rpoly\inparen{\vct{X}}$ and have defined it in a more ad-hoc manner, as suggested by the reviewers, including the comment immediately following.
@@ -168,13 +171,16 @@ We have added a reference. Please see \Cref{lem:val-ub}.
 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.}
+
+Thank you.  This reference was unnecessary and has been removed.
 
 \RCOMMENT{l.688 That sentence is hard to parse, consider reformulating it}
-\AH{Needs to be reformulated.}
+
+As the reviewer notes above, this section is unnecessary and we have removed it.
 
 \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.}
+
+Thank you.  The latter definition has been removed.
 
 
 \subsection{Reviewer 2}