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\section{Conclusions and Future Work}\label{sec:concl-future-work}

We have studied the problem of calculating the expectation of lineage 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.
% It has been studied extensively for sets (lineage formulas), but the bag settings has not received much attention.
While the expectation of a polynomial can be calculated in linear time for % in the size of
 polynomials % that are
in SOP form, the problem is \sharpwonehard for factorized polynomials (proven through a reduction from the problem of counting k-matchings).
%We have proven this claim through a reduction from the problem of counting k-matchings.
We prove that it is possible to approximate the expectation of a lineage polynomial in linear time
% When only considering polynomials for result tuples of
for UCQs over TIDBs and BIDBs (assuming that there are few cancellations).
Interesting directions for future work include development of a dichotomy for bag \abbrPDB\xplural.  While we handle higher moments in \Cref{app:sec-cicuits}, more general approximations are an interesting area for exploration, including those for more general 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:
% \textbullet{More queries: what happens with negation can circuits with monus be used?}
% \textbullet{More databases: can we push beyond BIDBs? E.g., C-tables / aggregate semimodules or just TIDBs where each input tuple is a random variable over $\mathbb{N}$?}
% \textbullet{Other results: can we extend the work to approximate $P(R(t) = n)$}
% }

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