ODIn/paper-BagRelationalPDBsAreHard
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Oliver Kennedy 2021-04-09 11:00:06 -04:00
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intro-new.tex
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@ -163,7 +163,7 @@ Concretely, we make the following contributions:
\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\Phi$. In fact, it turns out that for the TIDB (and BIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the TIDB/BIDB. Next, we motivate this reduced polynomial by continuing~\Cref{ex:intro-tbls}.
%Moving forward, we focus exclusively on bags.
Consider the query $Q():-$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$ $OnTime(\text{City}')$\OK{Should we be using RA- or Datalog-style query notation?} over the bag relations of \cref{fig:ex-shipping-simp}. It can be verified that $\Phi$ for $Q$ is $L_aL_b + L_bL_d + L_bL_c$. Now consider the product query $\poly^2():- Q \times Q$.
Consider the query $Q():-$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$ $OnTime(\text{City}')$ over the bag relations of \cref{fig:ex-shipping-simp}. It can be verified that $\Phi$ for $Q$ is $L_aL_b + L_bL_d + L_bL_c$. Now consider the product query $\poly^2():- Q \times Q$.
%The factorized representation of $\poly^2$ is (for simplicity we ignore the random variables of $Route$ since each variable has probability of $1$):
%\begin{equation*}
%\poly^2 = \left(L_aL_b + L_bL_d + L_bL_c\right) \cdot \left(L_aL_b + L_bL_d + L_bL_c\right)