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\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.
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.
% 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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