Update on Overleaf.

introduction.tex

@@ -142,16 +142,17 @@ Further, our approximation algorithm works for a more general notion of bag \abb
 \subsection{Polynomial Equivalence}\label{sec:intro-poly-equiv}
 A common encoding of probabilistic databases (e.g., in \cite{IL84a,4497507,DBLP:conf/vldb/AgrawalBSHNSW06} and many others) annotates tuples with lineages, propositional formulas that describe the set of possible worlds that the tuple appears in.  The bag semantics analog is a provenance/lineage polynomial (see~\Cref{fig:nxDBSemantics}) $\apolyqdt$~\cite{DBLP:conf/pods/GreenKT07}, a polynomial with non-zero integer coefficients and exponents, over variables $\vct{X}$ encoding input tuple multiplicities. The lineage polynomial for result tuple $t_{out}$ evaluates to $t_{out}$'s multiplicity in a given possible world when each $X_{t_{in}}$ is replaced by the multiplicity of $t_{in}$ in the possible world.
 
-We drop $\query$, $\tupset$, and $\tup$ from $\apolyqdt$ when they are clear from the context or irrelevant to the discussion. We now state the problem of computing the expectation of tuple multiplicity in terms of lineage polynomials (which is equivalent to \Cref{prob:bag-pdb-poly-expected}-- see \Cref{prop:expection-of-polynom}):
+ We now state the problem of computing the expectation of tuple multiplicity in terms of lineage polynomials (which is equivalent to \Cref{prob:bag-pdb-poly-expected}-- see \Cref{prop:expection-of-polynom}):
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Problem}[Expected Multiplicity of Lineage Polynomials]\label{prob:bag-pdb-poly-expected}
 Given an $\raPlus$ query $\query$, \abbrCTIDB $\pdb$ and result tuple $\tup$, 
 %compute the expected multiplicity of the polynomial $\poly$ (i.e., 
 %for $\worldvec\in\worlds$,
-compute $\expct_{\vct{W}\sim \pdassign}\pbox{\poly\inparen{\worldvec}}$).
+compute $\expct_{\vct{W}\sim \pdassign}\pbox{\poly\inparen[\query,\tupset,\tup]{\worldvec}}$).
 \end{Problem}
 %We note that computing \Cref{prob:expect-mult} is equivalent (yields the same result as) to computing \Cref{prob:bag-pdb-poly-expected} (see \Cref{prop:expection-of-polynom}).
 
+We drop $\query$, $\tupset$, and $\tup$ from $\apolyqdt$ when they are clear from the context or irrelevant to the discussion.
 All of our results rely on working with a {\em reduced} form $\inparen{\rpoly}$ of the lineage polynomial $\poly$. As we show, for the $1$-\abbrTIDB case, computing the expected multiplicity (over bag query semantics) is {\em exactly} the same as evaluating $\rpoly$ over the probabilities that define the $1$-\abbrTIDB.  
 Further, only light extensions (see \Cref{def:reduced-poly-one-bidb}) are required to support block independent disjoint probabilistic databases~\cite{DBLP:conf/icde/OlteanuHK10} (bag query semantics with input tuple multiplicity at most $1$). %, for which the proof of~\Cref{lem:tidb-reduce-poly} (introduced shortly) holds .