ra

intro-rewrite-070921.tex

@@ -245,7 +245,7 @@ We show in \Cref{sec:gen}
 A key insight of this paper is that the representation of $\circuit$ matters.
 For example, if we insist that $\circuit$ represent the lineage polynomial in the standard monomial basis (henceforth, \abbrSMB)\footnote{
   This is the representation, typically used in set-\abbrPDB\xplural, where the polynomial is reresented as sum of `pure' products. See \Cref{def:smb} for a formal definition.
-}, the answer to the above question in general is no, since then we will need $\abs{\circuit}\ge \Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^k}$, and hence, just $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ will be too large.
+}, the answer to the above question in general is no, since then we will need $\abs{\circuit}\ge \Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^k}$\BG{should be $|\idb |$?}, and hence, just $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ will be too large.
 
 However, systems can directly emit compact, factorized representations of $\poly(\vct{X})$ (e.g., as a consequence of the standard projection push-down optimization~\cite{DBLP:books/daglib/0020812}).
 For example, in~\Cref{fig:two-step}, $B(Y+Z)$ is a factorized representation of the SMB-form $BY+BZ$.

ra-to-poly.tex

@@ -38,7 +38,7 @@ For a probabilistic  database $\pdb = (\idb, \pd)$,  the result of a query is th
 Recall \Cref{fig:nxDBSemantics} which depicts the semantics for constructing a lineage polynomial $\apolyqdt$ for any $\raPlus$ query.  We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
 
 \begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
-Given a \abbrBPDB $\pdb = (\idb,\pd)$ and lineage polynomial $\apolyqdt$ for aribitrary output tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, 
+Given a \abbrBPDB $\pdb = (\idb,\pd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for aribitrary output tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
 we have (denoting $\randDB$ as the random variable over $\idb$):
   $ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
 \end{Proposition}