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38
app_notation-background.tex
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@ -20,7 +20,7 @@ To justify the use of $\semNX$-databases, we need to show that we can encode any
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As mentioned above we will use $\semNX$-databases paired with a probability distribution as a representation system, referring to such databases as \abbrNXPDB\xplural.
Formally, an \abbrNXPDB is an $\semNX$-database $\idb_{\semNX}$ and a probability distribution $\pd$ over assignments $\assign$ of the variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ occurring in annotations of $\idb_{\semNX}$ to $\{0,1\}$.
Formally, an \abbrNXPDB is an $\semNX$-database $\idb_{\semNX}$ and a probability distribution $\pd$ over assignments $\assign$ of the variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$ occurring in annotations of $\idb_{\semNX}$ to $\{0,1\}$.
\AH{There was a big ICDT reviewer complaint in this section, but I don't know that I think it confuses things to think of them both an assignment and/or a vector of variables.}
Note that an assignment $\assign: \vct{X} \to \{0,1\}^\numvar$ can be represented as a vector $\vct{w} \in \{0,1\}^n$ where $\vct{w}[i]$ records the value assigned to variable $X_i$. Thus, from now on we will solely use such vectors which we refer to as \emph{world vectors} and implicitly understand them to represent assignments. Given an assignment $\assign$ we use $\assign(\pxdb)$ to denote the semiring homomorphism $\semNX \to \semN$ that applies the assignment $\assign$ to all variables of a polynomial and evaluates the resulting expression in $\semN$.\BG{explain connection to homomorphism lifting in K-relations}
@ -51,7 +51,7 @@ Importantly, as the following proposition shows, any finite $\semN$-PDB can be e
%\subsection{Proof of~\Cref{prop:semnx-pdbs-are-a-}}
\begin{proof}
To prove that \abbrNXPDB\xplural are complete consider the following construction that for any $\semN$-PDB $\pdb = (\idb, \pd)$ produces an \abbrNXPDB $\pxdb = (\idb_{\semNX}, \pd')$ such that $\rmod(\pxdb) = \pdb$. Let $\idb = \{D_1, \ldots, D_{\abs{\idb}}\}$ and let $max(D_i)$
To prove that \abbrNXPDB\xplural are complete consider the following construction that for any $\semN$-PDB $\pdb = (\idb, \pd)$ produces an \abbrNXPDB $\pxdb = (\idb_{\semNX}, \pd')$ such that $\rmod(\pxdb) = \pdb$. Let $\idb = \{D_1, \ldots, D_{\abs{\idb}}\}$ and let $max(D_i)$
\AH{What are we using $max(D_i)$ for?}
denote $max_{\tup} D_i(\tup)$. For each world $D_i$ we create a corresponding variable $X_i$.
%variables $X_{i1}$, \ldots, $X_{im}$ where $m = max(D_i)$.
@ -61,7 +61,7 @@ In $\idb_{\semNX}$ we assign each tuple $\tup$ the polynomial:
\idb_{\semNX}(\tup) = \sum_{i=1}^{\abs{\idb}} D_i(\tup)\cdot X_{i}
\]
The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{w}_i$. All elements of $\vct{w}_i$ are zero except for the position corresponding to variables $X_{i}$ which is set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, $\semNX$ are a complete representation system.
It is trivial to show that an assignment $\vct{X} \to \{0,1\}$ is a semiring homomorphism.
Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}: \vct{X} \to \{0,1\}$ used here, uniquely extends to a semiring homomorphism $\textsc{Eval}_{\assign_{\vct{w}}}: \semNX \to \semN$. For a polynomial $\textsc{Eval}_{\assign_{\vct{w}}}(\poly)$ substitutes variables based on $\assign_{\vct{w}}$ and then evaluate the resulting expression in $\semN$. For instance, consider the polynomial $\poly = X + Y$ and assignment $\assign \defas X = 0, Y=1$. We get $\textsc_{\assign}(\poly) = 0 + 1 = 1$. % It is trivial to show that an assignment is a semiring homomorphism.
Closure under $\raPlus$ queries follows from this, and from \cite{DBLP:conf/pods/GreenKT07}'s Proposition 3.5, which states that semiring homomorphisms commute with queries over $\semK$-relations.
@ -91,25 +91,25 @@ 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.
As already noted above, in this work, we define \tis and \bis as subclasses of \abbrNXPDB\xplural.
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 $\vct{w}[j] = 1$).
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 $\vct{w}[j] = 1$).
For BIDBs specifically, note that at most one of the bits corresponding to tuples in each block will be set (i.e., for any pair of bits $w_j$, $w_{j'}$ that are part of the same block $b_i \supseteq \{t_{i,j}, t_{i,j'}\}$, at most one of them will be set).
Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$. Let $\pd^{(\vct{p})}$ denote the distribution induced by $\vct{p}$.
%
\begin{align}\label{eq:tidb-expectation}
\expct_{\vct{W} \sim \pd^{(\vct{p})}}\pbox{\poly(\vct{W})}
\expct_{\vct{W} \sim \pd^{(\vct{p})}}\pbox{\poly(\vct{W})}
= \sum\limits_{\substack{\vct{w} \in \{0, 1\}^\numvar\\ s.t. w_j,w_{j'} = 1 \rightarrow \not \exists b_i \supseteq \{t_{i,j}, t_{i',j}\}}} \poly(\vct{w})\prod_{\substack{j \in [\numvar]\\ s.t. \wElem_j = 1}}\prob_j \prod_{\substack{j \in [\numvar]\\s.t. w_j = 0}}\left(1 - \prob_i\right)
\end{align}
%
Recall that tuple blocks in a TIDB always have size 1, so the outer summation of \cref{eq:tidb-expectation} is over the full set of vectors.
\AH{Have cut and pasted the subsequent text. Need to verify this is the appropriate place for it.}
Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}.
We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}.
$\semNX$-databases are functions from tuples to elements of $\semNX$, typically called annotations.
Given an $\semNX$-database $\db$, it is common to use $\db(\tup)$ to denote the polynomial annotating tuple $\tup$ in $\db$.
%Note that based on this definition of $\rel$, $\rel(\tup)$ is the lineage polynomial for $\tup$.
Given an $\semNX$-database $\db$, it is common to use $\db(\tup)$ to denote the polynomial annotating tuple $\tup$ in $\db$.
%Note that based on this definition of $\rel$, $\rel(\tup)$ is the lineage polynomial for $\tup$.
Let $\numvar$ be the number of tuples in $\pdb$. Then, each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
The multiplicity of $\tup \in \db$, denoted $\db(\tup)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $\tup$ on $\vct{\wElem}$.
$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
@ -154,7 +154,7 @@ $\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} = \{d_1,\ldots, d_\numvar\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}%[Proof for~\Cref{lem:pre-poly-rpoly}]
Follows by the construction of $\rpoly$ in \cref{def:reduced-bi-poly}.
Follows by the construction of $\rpoly$ in \cref{def:reduced-bi-poly}.
\qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -171,10 +171,10 @@ $% \]
\end{Proposition}
\begin{proof}%[Proof for~\Cref{proposition:q-qtilde}]
Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion. For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$.
Finally, note that there are exactly three cases where the expectation of a monomial term $\expct\left[c_{\vct{d}}\prod_{i = n\; s.t.\; \vct{d}_i \geq 1}X_i\right]$ is zero:
(i) when $c_{\vct{d}} = 0$,
(ii) when $p_i = 0$ for some $i$ where $\vct{d}_i \geq 1$, and
Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion. For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$.
Finally, note that there are exactly three cases where the expectation of a monomial term $\expct\left[c_{\vct{d}}\prod_{i = n\; s.t.\; \vct{d}_i \geq 1}X_i\right]$ is zero:
(i) when $c_{\vct{d}} = 0$,
(ii) when $p_i = 0$ for some $i$ where $\vct{d}_i \geq 1$, and
(iii) when $X_i$ and $X_j$ are in the same block for some $i,j$ where $\vct{d}_i, \vct{d}_j \geq 1$.
\qed
\end{proof}
@ -207,4 +207,10 @@ Finally, it can be verified that \Cref{p1-s5} follows since \cref{p1-s4} satisfi
\begin{proof}
Note that \cref{lem:exp-poly-rpoly} shows that $\expct\pbox{\poly} =$ $\rpoly(\prob_1,\ldots, \prob_\numvar)$. Therefore, if $\poly$ is already in \abbrSMB form, one only needs to compute $\poly(\prob_1,\ldots, \prob_\numvar)$ ignoring exponent terms (note that such a polynomial is $\rpoly(\prob_1,\ldots, \prob_\numvar)$), which indeed has $\bigO{\size\inparen{\smbOf{\poly}}}$ computations.
\qed
\end{proof}
\end{proof}
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7
main.bib
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@ -640,3 +640,10 @@ Maximilian Schleich},
biburl = {https://dblp.org/rec/conf/pods/JoglekarPR16.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}
@book{graetzer-08-un,
author = {Gr{\"a}tzer, George},
title = {Universal algebra},
year = 2008,
publisher = {Springer Science \& Business Media}
}

47
rebuttal.tex
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@ -35,7 +35,7 @@ Our revision has removed the example referred to above. While the paper conside
\RCOMMENT{- In the case of set semantics, the lineage of a tuple can be defined for *any* query: it is the unique Boolean function that satisfies the if and only if property that you mention on line 70. For bag semantics however, to the best of my knowledge there is no general definition of what is a lineage for an arbitrary query. On line 73, it is not clear at all how the polynomial should be defined, since this will depend on the type of query that you consider}
The definition of the lineage polynomial (bag \abbrPDB) semantics over an arbitrary $\raPlus$ query $\query$ is modeled in \Cref{fig:nxDBSemantics}.
We also note that these semantics are not novel (e.g., similar semantics appear for both provenance \cite{DBLP:conf/pods/GreenKT07} and probabilistic database \cite{feng:2019:sigmod:uncertainty,kennedy:2010:icde:pip} contexts).
We also note that these semantics are not novel (e.g., similar semantics appear for both provenance \cite{DBLP:conf/pods/GreenKT07} and probabilistic database \cite{feng:2019:sigmod:uncertainty,kennedy:2010:icde:pip} contexts).
However, as we were likewise unable to find a formal proof of equivalence between the expectation of the query multiplicity and of the lineage polynomial, we prove it with \Cref{prop:expection-of-polynom}.
\RCOMMENT{l.75 "evaluating the lineage of t over an assignment corresponding to a possible world": here, does the assignment assigns each tuple to true or false? In other words, do the variables X still represent individual tuples? From what I see later in the article it seems that no, so this is confusing if we compare to what is explained in the previous paragraph about set TIDB}
@ -50,8 +50,8 @@ We have rewritten \Cref{sec:intro} in a way to stress that we are are primarily
\RCOMMENT{A discussion is missing about the difference between the approach usually taken in PDB literature and your approach. In which case would one be more interested in the expected multiplicity or in the marginal probability of a tuple? This should be discussed clearly in the introduction, as currently there is no clear "motivation" to what you do. There is a section about related work at the end but it is mostly a set of facts and there is no insightful comparison to what you do.}
We provide more motivating examples in the first paragraph, and include a more detailed discussion of the relationship to sets in paragraph \textbf{Relationship to Set-Probabilistic Query Evaluation} after \Cref{prob:informal}.
\AH{We need to maybe talk about the motivation for computing expected multiplicity.}
Broadly, expected multiplicities correspond to expected \lstinline{COUNT(*)} queries.
As a trivial (albeit relevant) example, consider a model of a contact network.
Broadly, expected multiplicities correspond to expected \lstinline{COUNT(*)} queries.
As a trivial (albeit relevant) example, consider a model of a contact network.
The probability that there exists at least one new COVID infection in the graph is far less informative than the expected number of new infections.
@ -68,14 +68,14 @@ The text now refers to latter as an \abbrNXPDB\xplural.
\RCOMMENT{If you want to be in the setting of bag PDBs, why not consider that the value of the variables are integers rather that Boolean? I.e., consider valuations $\nu: X \rightarrow$ N (or even to R, why not?) instead of $X \rightarrow \{0,1\}$; this would seem more natural to me than having this ad-hoc "mix" of Boolean and non-Boolean setting. If you consider this then your "reduced polynomial" trick does not seem to work anymore.}
Our objective is to establish the feasibility of deterministic-speed bag-probabilistic databases.
Our objective is to establish the feasibility of deterministic-speed bag-probabilistic databases.
Accordingly, we take our input model from production database systems like Postgresql, Oracle, DB2, SQLServer, etc\ldots (e.g., see \Cref{footnote:set-not-limit} on \Cpageref{footnote:set-not-limit}), where duplicate tuples are represented as independent entities.
As a convenient benefit, this leads to a direct translation of TIDBs (which are defined over 0,1 inputs).
Finally, as we mention earlier in the rebuttal, an easy generalization exists to encode a \abbrBPDB in a set-\abbrPDB (which then allows for bag inputs).
\RCOMMENT{- l.656 "Thus, from now on we will solely use such vectors...": this seems to
be false. Moreover you keep switching notation which makes it very hard to read... Sometimes it is $\varphi$, sometimes it is small w, sometimes it is big W (l.174 or l.722), sometimes the database is $\varphi(D)$, sometimes it is $\varphi_w(D)$, other times it is $D_{[w]}$ (l.671), and so on.}
We have made effort to be deliberately consistent with the use of notation, following standard usage whenever possible.
We have made effort to be deliberately consistent with the use of notation, following standard usage whenever possible.
\AH{We need to be sure this is taken care of in the appendix.}
\RCOMMENT{l.658 "we use $\varphi(D)$ to denote the semiring homomorphism $\semNX \rightarrow \semN$
@ -96,12 +96,16 @@ The semantics for the polynomial as seen in \Cref{eq:sop-form} is specified inde
\RCOMMENT{Proof of Proposition A.3. I seems the proof should end after l.687, since you already proved everything from the statement of the proposition. I don't understand what it is that you do after this line.}
This text is an informal proof of \Cref{prop:expection-of-polynom} originally intended to motivate \Cref{prop:semnx-pdbs-are-a-}.
This text is an informal proof of \Cref{prop:expection-of-polynom} originally intended to motivate \Cref{prop:semnx-pdbs-are-a-}.
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.}
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.}
@ -118,8 +122,8 @@ Based on this and other reviewer comments, we removed the formal definition of $
As alluded to above, we have incorporated the reviewer's suggestion, c.f. \Cref{def:reduced-poly} and \Cref{def:reduced-bi-poly}.
\RCOMMENT{Definition 2.14. It is not clear what is the input exactly. Are the query Q and database D fixed? Moreover, I have the impression that your hardness results have nothing to do with lineages and that you don't need them to express your results. I think the problem you should consider is simply the following: Expected Multiplicity Problem: Input: query Q, N[X]-database D, tuple t. Output: expected multiplicity of t in Q(D). Your main hardness result would then look like this: the Expected
Multiplicity problem restricted to conjunctive queries is \#W[1]-hard, parameterized by query size. Indeed if I look at the proof, all you need is the queries $Q^k_G$. The problem is \#W[1]-hard and it should not matter how one tries to solve it: using an approach with lineages or using anything else.
Multiplicity problem restricted to conjunctive queries is \#W[1]-hard, parameterized by query size. Indeed if I look at the proof, all you need is the queries $Q^k_G$. The problem is \#W[1]-hard and it should not matter how one tries to solve it: using an approach with lineages or using anything else.
Currently it is confusing because you make it look like the problem is hard only when you consider general arithmetic circuits, but your hardness proof has nothing to do with circuits. Moreover, it is not surprising that computing the expected output of an arithmetic circuit is hard: it is trivial, given a CNF $\phi$, to build an arithmetic circuit C such that for any valuation $\nu$ of the variables the formula $\phi$ evaluates to True under $\nu$ if C evaluates to 1 and the formula $\phi$ evaluates to False under $\nu$ if C evaluates to 0, so this problem is \sharpphard anyways.}
We have rewritten \Cref{sec:intro} with a series of refined problem statements to show that the problem we explore and the results we obtain directly involve lineage polynomials. The reviewer is correct that the output is the expected multiplicity, and we hope that our updated presentation of the paper makes it clear that $\expct_{\vct{\randWorld}\sim\pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ is indeed the expected multiplicity spoken of. We have also addressed the ambiguity in the complexity we are focusing on, both explicitly in the intro and in the revised definition, \Cref{def:the-expected-multipl}.
@ -146,7 +150,7 @@ More specifically, our proofs rely on (i) circuits with a bounded polynomial deg
\RCOMMENT{l.411: what are $|C|^2(1,...,1)$ and $|C|(1,...,1)$? }
We clarify this overloaded notation immediately after \Cref{def:positive-circuit}.
\RCOMMENT{Sometimes you consider UCQs, sometimes RA+ queries. I think it would be simpler if you stick to one formalism (probably UCQs is cleaner?)}
\RCOMMENT{Sometimes you consider UCQs, sometimes RA+ queries. I think it would be simpler if you stick to one formalism (probably UCQs is cleaner?)}
As alluded to previously, we have followed the reviewer's suggestion and have found $\raPlus$ queries to be most amenable for this work.
\RCOMMENT{l.432 what is an FAQ query?}
@ -157,13 +161,13 @@ 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. This observation has persuaded us to eliminate $\semNX$-DB query evaluation and have only an algorithm for lineage.
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.}
\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.}
@ -188,9 +192,9 @@ We have fixed this mistake.
\RCOMMENT{Definition 2.7. "valid worlds $\eta$". This is confusing. A "possible world" is an element of $\idb$: this is not stated explicitly in the paper, but it is implicit on line 163, so I assumed that possible worlds refer to elements of $\idb$. If I assumed correctly, then calling $\eta$ a "world" in Def. 2.7 is misleading, because $\eta$ is not an element of $\idb$. More, it is unclear to me why this definition is needed: it is used right below, in Lemma 2.8, but that lemma seems to continue to hold even if w is not restricted.}
We agree with the reviewer that this notation is confusing;
We agree with the reviewer that this notation is confusing;
$\eta$ is meant to cope with the fact that tuples from the same group in a BIDB can not co-exist, even though our $\{0,1\}$-input vectors can encode such worlds.
We now address this constraint by embedding it directly into the reduced polynomial with \Cref{def:reduced-bi-poly}.
We now address this constraint by embedding it directly into the reduced polynomial with \Cref{def:reduced-bi-poly}.
\AH{Needs to be addressed.}
\OK{We have addressed this... but may still need to elide $\eta$ out of the appendices}
@ -226,10 +230,10 @@ Our revision has eliminated this statement.
\RCOMMENT{The coverage of related work is adequate. Fink et. al seems as the closest related work to me and I would appreciate a more elaborate comparison with this paper. My understanding is that Fink et. al considers exact evaluation only and focuses on knowledge compilation techniques based on decompositions. They also note that "Expected values can lead to unintuitive query answers, for instance when data values and their probabilities follow skewed and non-aligned distributions" attributed to [2]. Does this apply to the current work? Can you please comment on this?}
The work is indeed quite close to our own.
It targets a broader class of queries (aggregates include COUNT/SUM/MIN/MAX, rather than our more narrow focus on COUNT), but has significantly less general theoretical results.
Most notably, their proof of linear runtime in the size of the input polynomial is based on a tree-based encoding of the polynomial.
Tree-based representation representation (and hence the Fink et. al. algorithm's runtime) is, as we note several times, superlinear in $\qruntime{\query, \dbbase}$.
This result is also limited to a specific class of (hierarchical) queries, devolving to exponential time (as in \cite{FH13}) in general.
It targets a broader class of queries (aggregates include COUNT/SUM/MIN/MAX, rather than our more narrow focus on COUNT), but has significantly less general theoretical results.
Most notably, their proof of linear runtime in the size of the input polynomial is based on a tree-based encoding of the polynomial.
Tree-based representation representation (and hence the Fink et. al. algorithm's runtime) is, as we note several times, superlinear in $\qruntime{\query, \dbbase}$.
This result is also limited to a specific class of (hierarchical) queries, devolving to exponential time (as in \cite{FH13}) in general.
By contrast, our results apply to all of $\raPlus$.
Our revised related work section now addresses both points.
@ -246,4 +250,11 @@ We have (as noted throughout this section) revised the writing to provide precis
\RCOMMENT{}
\RCOMMENT{}
\RCOMMENT{}
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