diff --git a/intro-new.tex b/intro-new.tex index d8f7fcb..3d5024c 100644 --- a/intro-new.tex +++ b/intro-new.tex @@ -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)