ODIn/paper-BagRelationalPDBsAreHard
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Atri Rudra 2020-12-19 23:44:40 -05:00
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conclusions.tex
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%!TEX root=./main.tex
\section{Conclusions and Future Work}\label{sec:concl-future-work}
We have studied the problem of calculating the expectation of polynomials over random integer variables.
We have studied the problem of calculating the expectation of query polynomials over BIDBs. %random integer variables.
This problem has a practical application in probabilistic databases over multisets, where it corresponds to calculating the expected multiplicity of a query result tuple.
This problem has been studied extensively for sets (lineage formulas), but the bag settings has not received much attention so far.
While the expectation of a polynomial can be calculated in linear time in the size of polynomials that are in SOP form, the problem is \sharpwonehard for factorized polynomials.
We have proven this claim through a reduction from the problem of counting k-matchings.
When only considering polynomials for result tuples of UCQs over TIDBs and BIDBs (under the assumption that there are $O(1)$ cancellations), we prove that it is still possible to approximate the expectation of a polynomial in linear time.
An interesting direction for future work would be development of a dichotomy for queries over bag PDBs.
When only considering polynomials for result tuples of UCQs over TIDBs and BIDBs (under the assumption that there are few cancellations), we prove that it is still possible to approximate the expectation of a polynomial in linear time.
Interesting directions for future work include development of a dichotomy for queries over bag PDBs and desgin approximation schemes for data models beyond what we consider in this paper.
% Furthermore, it would be interesting to see whether our approximation algorithm can be extended to support queries with negations, perhaps using circuits with monus as a representation system.
\BG{I am not sure what interesting future work is here. Some wild guesses, if anybody agrees I'll try to flesh them out:

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related-work.tex
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%!TEX root=./main.tex
\section{Related Work}\label{sec:related-work}
In addition to probabilistic databases, our work has connections to work on compact representations of polynomials and on fine-grained complexity which we review in \Cref{sec:compr-repr-polyn,sec:param-compl}.
In addition to probabilistic databases, our work has connections to work on compact representations of polynomials and on fine-grained complexity, which we review in \Cref{sec:compr-repr-polyn,sec:param-compl}.
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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
@ -19,7 +19,7 @@ This is similar to our $\semNX$-PDBs, but we use polynomials instead of Boolean
Approaches for probabilistic query processing (i.e., computing the marginal probability for query result tuples), fall into two broad categories.
\emph{Intensional} (or \emph{grounded}) query evaluation computes the \emph{lineage} of a tuple % (a Boolean formula encoding the provenance of the tuple)
and then the probability of the lineage formula.
In this paper we focus on intensional query evaluation using polynomials instead of boolean formulas.
In this paper we focus on intensional query evaluation using polynomials instead of Boolean formulas.
It is a well-known fact that computing the marginal probability of a tuple is \sharpphard (proven through a reduction from weighted model counting~\cite{valiant-79-cenrp} %provan-83-ccccptg
using the fact the tuple's marginal probability is the probability of a its lineage formula).
The second category, \emph{extensional} query evaluation, % avoids calculating the lineage.