Merge branch 'master' of gitlab.odin.cse.buffalo.edu:ahuber/SketchingWorlds

app_notation-background.tex

@@ -91,7 +91,7 @@ Two important subclasses of \abbrNXPDB\xplural that are of interest to us are th
 The probability of such a world is the product of the probabilities of all tuples present in the world.  %and one minus the sum of the probabilities of all tuples from blocks for which no  tuple is present in the world.
 For bag \tis and \bis, we define the probability of a tuple to  be the probability that the tuple exists with multiplicity at least $1$.
 
-As already noted above, in this work, we define \tis and \bis as subclasses of \abbrNXPDB\xplural.
+In this work, we define \tis and \bis as subclasses of \abbrNXPDB\xplural defined over variables $\vct{X}$ (\Cref{def:semnx-pdbs}) where $\vct{X}$ can be partitioned into blocks that satisfy the conditions of a \ti or \bi (stated formally in \Cref{subsec:tidbs-and-bidbs}).
 In this work, we consider one further deviation from the standard: We use bag semantics for queries.
 Even though tuples cannot occur more than once in the input \ti or \bi, they can occur with a multiplicity larger than one in the result of a query.
 Since in \tis and \bis, there is a one-to-one correspondence between tuples in the database and variables, we can interpret a vector $\vct{w} \in \{0,1\}^n$ as denoting which tuples exist in the possible world $\assign_{\vct{w}}(\pxdb)$ (the ones where $\vct{w}[j] = 1$).

rebuttal.tex

@@ -100,14 +100,18 @@ This text is an informal proof of \Cref{prop:expection-of-polynom} originally in
 We agree that this should not be part of the proof of the later, and have removed the text.
 
 \RCOMMENT{l.686 "The closure of ... over K-relations": you should give more details on this part. It is not obvious to me that the relations from l.646 hold.}
-\AH{This too needs to be looked at.}
+
+The core of this (otherwise trivial) argument, that semiring homomorphisms commute through queries, was already proven in \cite{DBLP:conf/pods/GreenKT07}.  We now make this reference explicit. 
+
 
 We apologize for not explaining this in more detail. In universal algebra~\cite{graetzer-08-un}, it has been proven (the HSP theorem) that for any variety, the set of all structures (called objects) with a certain signature that obey a set of equational laws, there exists a ``most general'' object called the \emph{free object}. The elements of the free objects are equivalence classes (with respect to the laws of the variety) of symbolic expressions over a set of variables $\vct{X}$ that consist of the operations of the structure. The operations of the free object are combining symbolic expression using the operation. It has been shown that for any other object $K$ of a variety, any assignment $\phi: \vct{X} \to K$ uniquely extends to a homomorphism from the free object to $K$ by substituting variables for based on $\phi$ in symbolic expression and then evaluating the resulting expression in $K$.
 
 Commutative semirings form a variety where $\semNX$ is the free object. Thus, for any polynomial (element of $\semNX$), for any assignment  $\phi: \vct{X} \to \semN$ (also a semiring) there exists a unique semiring homomorphism $\textsc{Eval}_{\phi}: \semNX \to \semN$. Homomorphisms by definition commute with the operations of a semiring. Green et al. \cite{GK07} did prove that semiring homomorphisms extend to homomorphisms over K-relations (by applying the homomorphism to each tuple's annotation) and these homomorphisms over K-relations commute with queries.
 
 \RCOMMENT{l.711 "As already noted...": ah? I don't see where you define which subclass of N[X]-PDBs define bag version of TIDBs. If this is supposed to be in Section 2.1.1 this is not clear, since the world "bag" does not even appear there (and as already mentioned everything seems to be set semantics in this section). I fact, nowhere in the article can I see a definition of what are bag TIDBs/BIDBs}
-\AH{This needs to be taken care of in the appendix.}
+
+The new text precisely defines TIDBs (\Cref{sec:intro}), and the BIDB generalization (\Cref{subsec:tidbs-and-bidbs}).  The specific text referenced in this comment has now been restructured to reference \Cref{def:semnx-pdbs} (which defines an \abbrNXPDB defined over variables $\vct{X}$) and relate it to the formal structure of BIDBs in \Cref{subsec:tidbs-and-bidbs}.
+
 
 \RCOMMENT{- l.707 "the sum of the probabilities of all the tuples in the same block b is  1": no, traditionally it can be less than 1, which means that there could be no tuple in the block.}
 The reviewer is correct and we have updated our appendix text accordingly.