Small changes.

intro-new.tex

@@ -134,7 +134,7 @@ Thus, the marginal probability of tuple $\tup$ is equal to the probability that
 
 For bag semantics, the lineage of a tuple is a polynomial over variables $\vct{X}=(X_1,\dots,X_n)$ with % \in \mathbb{N}^\numvar$ with
 coefficients in the set of natural numbers $\mathbb{N}$ (an element of semiring $\mathbb{N}[\vct{X}]$).
-Analogously to sets, evaluating the lineage for $t$  over an assignment corresponding to a possible world yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \Cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which we will denote as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
+Analogously to sets, evaluating the lineage for $t$  over an assignment corresponding to a possible world yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \Cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which for this example we denote\footnote{In later sections we will simply refer to $\linsett{\query}{\pdb}{\tup}$ as $Q$.} as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Example}\label{ex:intro-lineage}
@@ -159,7 +159,7 @@ Concretely, we make the following contributions:
 (i) We show that the expected result multiplicity problem for conjunctive queries for bag-$\ti$s is \sharpwonehard in the size of a lineage circuit by reduction from counting the number of $k$-matchings over an arbitrary graph;
 (ii) We present an $(1\pm\epsilon)$-\emph{multiplicative} approximation algorithm for bag-$\ti$s and show that for typical database usage patterns (e.g. when the circuit is a tree or is generated by recent worst-case optimal join algorithms or their FAQ followups~\cite{DBLP:conf/pods/KhamisNR16}) its complexity is linear in the size of the compressed lineage encoding; %;\BG{Fix not linear in all cases, restate after 4 is done}
 (iii) We generalize the approximation algorithm to bag-$\bi$s, a more general model of probabilistic data;
-(iv) We further prove that for \raPlus queries (a equivalently expressive, but factorizable form of UCQs),  we can approximate the expected output tuple multiplicities with only $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to higher moments of the tuple multiplicity (instead of just the expectation).
+(iv) We further prove that for \raPlus queries (an equivalently expressive, but factorizable form of UCQs),  we can approximate the expected output tuple multiplicities with only $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to higher moments of the tuple multiplicity (instead of just the expectation).
 
 %\mypar{Implications of our Results} As mentioned above