Added another comment.

intro.tex

@@ -11,9 +11,9 @@ Most implementations of modern probabilistic databases (PDBs) view the annotatio
 (Line 4): In this work we show that if we use better representation like factorized DBs for annotation polynomial then the complexity landscape becomes much more nuanced.
 }
 
-In practice, modern production databases, e.g., Postgres, Oracle, etc. use bag semantics.  In contrast, as noted above, most implementations of PDBs are built in the setting of set semantics,\AR{The stuff so far with minor modifications can function as the first two lines of the first para.} and this contributes to slow computation time \AR{Why did you put this comment on slow computation time? What is it buying you? It seems like you are jumping ahead over here.}. In both settings it is the case that each tuple is annotated with a polynomial, which describes the tuples contributing to the given tuple's presence in the output.  While in set semantics, the output polynomial is predominantly viewed as the probability\AR{The annotation polynomial does {\bf NOT} give the probability of a tuple-- it just says whether a tuple is present in a world or not. Only when you take the expecation of the polynomial do you get a probability value.} that its associated tuple exists or not, in the bags setting the polynomial  is an encoding of the multiplicity of the tuple in the output.  Note that in general, as we allude later one, the polynomial can also represent set semantics, access levels, and other encodings.\AR{How does the last statement help? In this paper we are interested in set and bag semantics: why are you bring up other possibilities that have nothing to do with the main message of the paper?} \AR{I'll pause here not since it would be quicker to give comments during the meeting but here is something to ponder about, which might make things simpler to present. I would recommend that you introduce the theoretical problem {\bf right after} the first para. Once you define the problem, what you are trying to describe in words in this para can be done much more succinctly with relevant notation. If you want to be bit more gentle then perhaps start off with an example annotation polynomial and talk about that example-- this latter one might be better for PODS. But {\em ideally}, you would like an example query that is hard in set semantics but easy in bag in SoP representation but also hard with more succinct representation. Side Q-- is our hard query that we use for triangle counting etc. \#P-hard in the set semantics? If so that would be a great example to use throughout the intro.}  In bag semantics, the polynomial is composed of $+$ and $\times$ operators, with constants from the set $\mathbb{N}$ and variables from the set of variables $\vct{X}$.  Should we attempt to make computations, e.g. expectation, over the output polynomial, the naive algorithm cannot hope to do better than linear time in the size of the polynomial.  However, in the set semantics setting, when e.g., computing the expectation (probability) of the output polynomial given values for each variable in the polynomial's set of variables $\vct{X}$, this problem is \#P-hard. %of the output polynomial of a result tuple for an arbitrary query.  In contrast, in the bag setting, one cannot generate a result better than linear time in the size of the polynomial.
+In practice, modern production databases, e.g., Postgres, Oracle, etc. use bag semantics.  In contrast, as noted above, most implementations of PDBs are built in the setting of set semantics,\AR{The stuff so far with minor modifications can function as the first two lines of the first para.} and this contributes to slow computation time \AR{Why did you put this comment on slow computation time? What is it buying you? It seems like you are jumping ahead over here.}. In both settings it is the case that each tuple is annotated with a polynomial, which describes the tuples contributing to the given tuple's presence in the output.  While in set semantics, the output polynomial is predominantly viewed as the probability\AR{The annotation polynomial does {\bf NOT} give the probability of a tuple-- it just says whether a tuple is present in a world or not. Only when you take the expecation of the polynomial do you get a probability value.} that its associated tuple exists or not, in the bags setting the polynomial  is an encoding of the multiplicity of the tuple in the output.  Note that in general, as we allude later one, the polynomial can also represent set semantics, access levels, and other encodings.\AR{How does the last statement help? In this paper we are interested in set and bag semantics: why are you bring up other possibilities that have nothing to do with the main message of the paper?} \AR{I'll pause here now (except for another comment on next page) since it would be quicker to give comments during the meeting but here is something to ponder about, which might make things simpler to present. I would recommend that you introduce the theoretical problem {\bf right after} the first para. Once you define the problem, what you are trying to describe in words in this para can be done much more succinctly with relevant notation. If you want to be bit more gentle then perhaps start off with an example annotation polynomial and talk about that example-- this latter one might be better for PODS. But {\em ideally}, you would like an example query that is hard in set semantics but easy in bag in SoP representation but also hard with more succinct representation. Side Q-- is our hard query that we use for triangle counting etc. \#P-hard in the set semantics? If so that would be a great example to use throughout the intro.}  In bag semantics, the polynomial is composed of $+$ and $\times$ operators, with constants from the set $\mathbb{N}$ and variables from the set of variables $\vct{X}$.  Should we attempt to make computations, e.g. expectation, over the output polynomial, the naive algorithm cannot hope to do better than linear time in the size of the polynomial.  However, in the set semantics setting, when e.g., computing the expectation (probability) of the output polynomial given values for each variable in the polynomial's set of variables $\vct{X}$, this problem is \#P-hard. %of the output polynomial of a result tuple for an arbitrary query.  In contrast, in the bag setting, one cannot generate a result better than linear time in the size of the polynomial.
 
-There is limited work and results in the area of bag semantic PDBs.  This work seeks to leverage prior work in factorized databases (e.g. Olteanu et. al.)~\cite{DBLP:conf/tapp/Zavodny11} with PDB implementations to improve efficient computation over output polynomials, with theoretical guarantees.  When considering PDBs in the bag setting a subtelty arises that is easily overlooked due to the \textit{oversimplification} of PDBs in the set setting, i.e., in set semantics expectation doesn't have linearity over disjunction, and a consequence of this is that it is not true in the general case that a compressed polynomial has an equivalent expectation to its DNF form.  In the bag PDB setting, however, expectation does enjoy linearity over addition, and the expectation of a compressed polynomial and its equivalent SOP are indeed the same.
+There is limited work and results in the area of bag semantic PDBs.  This work seeks to leverage prior work in factorized databases (e.g. Olteanu et. al.)~\cite{DBLP:conf/tapp/Zavodny11} with PDB implementations to improve efficient computation over output polynomials, with theoretical guarantees.  \AR{I know what you are trying to say in the rest of the para but it can be easily interpreted to be saying something that is {\bf false}. It is always the case that the compressed form of the polynomial always evaluates to the same value as the extended SoP form for any value. So the expcted value of compressed poly is {\em always the same} as expected value of the SoP forms. What you are trying to get here if when you "push" in the expectations. Again the latter is very hard to describe in words. But this would be much easier to state once you have the notation in place. Or if you have a runnign example.} When considering PDBs in the bag setting a subtelty arises that is easily overlooked due to the \textit{oversimplification} of PDBs in the set setting, i.e., in set semantics expectation doesn't have linearity over disjunction, and a consequence of this is that it is not true in the general case that a compressed polynomial has an equivalent expectation to its DNF form.  In the bag PDB setting, however, expectation does enjoy linearity over addition, and the expectation of a compressed polynomial and its equivalent SOP are indeed the same.
 
 For almost all modern PDB implementations, an output polynomial is only ever considered in its expanded SOP form.
 \BG{I don't think this statement holds up to scrutiny. For instance, ProvSQL uses circuits.}