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

We have studied the problem of calculating the expected multiplicity of a query result tuple, %expectation of lineage polynomials over BIDBs. %random integer variables.
 a problem that 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 show that under various parameterized complexity hardness results/conjectures computing the expected multiplicities exactly is not possible in time linear in the corresponding deterministic query processing time.
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
 in the deterministic query processing  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 can handle higher moments (this follows fairly easily from our existing results-- see \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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