Added proof to Prop 2.4; added text speaking of Grohe's work; started cleaning the appendix.
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@ -70,11 +70,11 @@ In this work, we consider one further deviation from the standard: We use bag se
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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.
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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 $w_i = 1$).
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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).
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Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$. Given \abbrPDB $\pdb$t $\pd$ is the distribution induced by $\vct{p}$, which we will denote $\pd^{\inparen{\vct{\prob}}}$.
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Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$. Given \abbrPDB $\pdb$ where $\pd$ is the distribution induced by $\vct{p}$, which we will denote $\pd^{\inparen{\vct{\prob}}}$.
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%
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\begin{align}\label{eq:tidb-expectation}
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\expct_{\vct{W} \sim \pd^{(\vct{p})}}\pbox{\poly(\vct{W})}
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= \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)
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= \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{i\in\pbox{\numblock}~|~\not\exists j\in [\numvar]\\s.t. w_j = 1}}\left(1 - \sum_{j\in\pbox{\numvar}~|~\tup_{i, j} \subseteq\block_i}\prob_i\right)
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\end{align}
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%
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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.
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@ -112,7 +112,7 @@ $% \]
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\begin{proof}
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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 $e \geq 1$, $b^e = b$.
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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:
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Finally, note that there are exactly three cases where the expectation of a monomial term $\expct\left[c_{\vct{d}}\prod_{i = 1\; s.t.\; \vct{d}_i \geq 1}^\numvar X_i\right]$ is zero:
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(i) when $c_{\vct{d}} = 0$,
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(ii) when $p_i = 0$ for some $i$ where $\vct{d}_i \geq 1$, and
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(iii) when $X_i$ and $X_j$ are in the same block for some $i,j$ where $\vct{d}_i, \vct{d}_j \geq 1$.
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@ -1,9 +1,15 @@
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%!TEX root=./main.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\input{app_set-to-bag-pdb}
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%%%%%%%%%%%%%%%%%%
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%Appendix A is no longer needed %
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%%%%%%%%%%%%%%%%%%
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%\input{app_set-to-bag-pdb}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\onecolumn
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\section{Missing details from Section~\ref{sec:background}}\label{sec:proofs-background}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -53,11 +53,17 @@ We slightly abuse notation here, denoting a world vector as $W$ rather than $\wo
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\Cref{fig:lin-poly-bidb} shows the lineage construction of $\poly'\inparen{\vct{X}}$ given $\raPlus$ query $\query$ for arbitrary deterministic $\gentupset'$. Note that the semantics differ from~\Cref{fig:nxDBSemantics} only in the base case.
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\begin{Proposition}[\abbrCTIDB reduction]\label{prop:ctidb-reduct}
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Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$ be the \emph{\abbrOneBIDB} obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$.
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Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$ be the \emph{\abbrOneBIDB} obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intuple{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$.
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The probability distribution $\bpd'$ is the characterized by the vector $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$.
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Then, the distributions $\mathcal{P}$ and $\mathcal{P}'$ are equivalent.
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\end{Proposition}
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{\color{red}
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\begin{proof}[Proof of~\Cref{prop:ctidb-reduct}]
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The distribution $\bpd$ is a distribution disjoint across the set of multiplicities $\pbox{\bound}$ and independent across all $\tup\in\tupset$. By definition,~\Cref{prop:ctidb-reduct} creates $\pdb'$ by producing a block of disjoint tuples $\tup_j = \intuple{\tup, j}$ for each $\tup\in\tupset$ and $j\in\pbox{\bound}$, where $\vct{\prob} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$. Since the probability vector $\vct{\prob_{\pdb}} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$ \emph{and} each $\prob_{\tup, j}, \prob_{\tup, j'}$ for $j\neq j'\in\pbox{\bound}$ are disjoint, the distributions are hence the same.
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\end{proof}
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}
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We now define the reduced polynomial $\rpoly'$ of a \abbrOneBIDB.
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\begin{figure}[t!]
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%\centering
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@ -259,6 +259,9 @@ Although \abbrPDB queries instead convey the trustworthiness of results~\cite{ku
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Bags, as we consider, are sufficient for production use, where bag-relational algebra is already the default for performance reasons.
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Our results show that bag-\abbrPDB\xplural can be competitive, laying the groundwork for probabilistic functionality in production database engines.
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\mypar{Concurrent Work}
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In work unpublished and independent of ours, Grohe, et. al. investigate \abbrBPDB\xplural in the context of infinite multiplicities, and show a dichotomy for the query evaluation problem over \abbrBPDB\xplural when computing the probability of an output tuple having at most a multiplicity of $k$. Their work, however, does not take into consideration the fine-grained analysis of computing the expected multiplicity of an output tuple.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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10
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@ -456,13 +456,17 @@
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\newblock
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\bibitem[pdbench utility([n.\,d.])]%
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\bibitem[pdbench utility(2008)]%
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{pdbench}
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pdbench utility \bibinfo{year}{[n.\,d.]}\natexlab{}.
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pdbench utility \bibinfo{year}{2008}\natexlab{}.
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\newblock \bibinfo{title}{pdbench}.
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\newblock \bibinfo{howpublished}{\url{http://pdbench.sourceforge.net/}}.
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\newblock
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\newblock
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\urldef\tempurl%
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\url{http://pdbench.sourceforge.net/}
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\showURL{%
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\tempurl}
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\newblock
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\shownote{Accessed: 2020-12-15}.
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2
main.bib
2
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@ -40,7 +40,7 @@ series = {FOCS '02}
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title = "pdbench",
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url = {http://pdbench.sourceforge.net/},
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note = {Accessed: 2020-12-15},
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year="2020"
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year="2008"
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}
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@article{AF18,
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77
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@ -18,43 +18,40 @@
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