diff --git a/k-relations.tex b/k-relations.tex
index 2d9aa57..361a593 100644
--- a/k-relations.tex
+++ b/k-relations.tex
@@ -1,20 +1,12 @@
 %!TEX root=./main.tex
-We can use $\domK$-relations to model bags. A \emph{$\domK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$.  A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
+We can use $\semK$-relations to model bags. A \emph{$\semK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$.  A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
 Let $\udom$ be a countable domain of values.
 Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$.
 A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.
 
-For completeness, we briefly review the semantics for $\raPlus$ queries over $\semK$-relations~\cite{DBLP:conf/pods/GreenKT07} illustrated in \Cref{fig:nxDBSemantics}.
-In \Cref{fig:nxDBSemantics}, we use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$, assume that tuples are of appropriate arity, use $\sch(\rel)$ to denote the attributes of $\rel$, and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$.  Furthermore, $\theta(\tup)$ denotes the (Boolean) result of evaluating condition $\theta$ over $\tup$.
+The semantics for $\raPlus$ queries over $\semK$-relations are analogous to the lineage construction semantics of \Cref{fig:nxDBSemantics}, with the exception of replacing $+$ with $\addK$ and $\cdot$ with $\multK$.
 
-Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A  probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases.  Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result tuple $t$,  we use $\query(\db)(t)$ for $\db \in \idb$ as input in RHS of \Cref{eq:intro-bag-expectation} to compute the expected multiplicity of $t$.  Note that the tables of \Cref{fig:ex-shipping-simp} have an implicit $1$ $\semN$-valued annotation for each tuple in tables $OnTime$ and $Route$.
-%\Cref{ex:intro-tbls} and \Cref{ex:intro-lineage} $\semN$-valued variable and are interested in computing its expectation  $\expct_{\idb \sim \probDist}[\query(\db)(t)]$:
-%%
-%\begin{equation}\label{eq:bag-expectation}
-%\expct_{\idb \sim \probDist}[\query(\db)(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \probOf(\db)
-%\end{equation}
-%%
-Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of $t$ we expect to find in result of query $\query$.
+Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A  probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases.  Given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, ($\raPlus$) query $\query$, and possible result tuple $\tup$, we sum $\query(\db)(\tup)\cdot\pd\inparen{\db}$ for all $\db \in \idb$ to compute the expected multiplicity of $\tup$.  Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of $t$ we expect to find in result of query $\query$.
 
 Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_n)$ with natural number coefficients and exponents.
 Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$, with the standard addition and multiplication of polynomials.