Done with Sec 1.1

ra-to-poly.tex

@@ -10,7 +10,7 @@
 \subsection{Intro}
 \AR{A funny story: we wrote a paper once where the first section was titled ``Intro"-- and no reviewers noticed it! But you should expand it out to Introduction}
 
-An incomplete database $\idb$ is a set of deterministic databases $\db_i$ where each element is known as a possible world.  Since $\idb$ is modeling all the possible worlds of an uncertain database, it follows that each $\db_i \in \idb$ has the same named set of relations, $\{\rel_1,\ldots, \rel_n\}$ (albeit not equivalent across all instances), whose schemas are unchanging across each $\db_i$.  When $\idb$ is a probabilistice database, $\idb$ can be viewed as having two components, the set of possible worlds, and a probability space $\left(\Omega, \mathcal{A}, P\right)$ over that set.  Since the set of possible outcomes is the set of possible worlds, $\wSet$, and the set of outcomes is equivalent to the set of events, we will simplify notation and use $\left(\wSet, P\right)$ to denote the probability space of $\idb$. 
+An incomplete database $\idb$ is a set of deterministic databases $\db_i$ where each element is known as a possible world.  Since $\idb$ is modeling all the possible worlds of an uncertain database, it follows that each $\db_i \in \idb$ has the same named set of relations, $\{\rel_1,\ldots, \rel_n\}$ (albeit not equivalent across all instances), whose schemas are unchanging across each $\db_i$.  When $\idb$ is a probabilistice database, $\idb$ can be viewed as having two components, the set of possible worlds, and a probability space $\left(\Omega, \mathcal{A}, P\right)$ over that set. \AR{I'm not sure why you are using the notation $\mathcal{A}$ and $P$, which you do not seem to use beyond this section. I would recommend that you only introduce a notation if you plan to use them later on.} Since the set of possible outcomes is the set of possible worlds, $\wSet$, and the set of outcomes is equivalent to the set of events, we will simplify notation and use $\left(\wSet, P\right)$ to denote the probability space of $\idb$. \AR{If you want to use $(\wSet,P)$ make sure you use the same notation in Sec 1.3 as well. If not, then use the notation from Sec 1.3 here}
 \subsection{Modeling and Semantics}
 $\idb$ can be generally viewed as the set of relations $\{\prel_1,\ldots, \prel_n\}$, where for each $\prel_i \in \idb$, $\prel_i$ consists of the set of all tuples appearing in $\rel_i$ across each of the possible worlds $\db_i \in \idb$, where each tuple is annotated with a provenance polynomial from the set $\mathbb{N}[X]$, and the set $X$ is the alphabet of variables in $\idb$.  One can think of $\idb$ as a parameterized database, whose abstract form maps to a deterministic $\db_i \in \idb$ based on the valuation to which the variables of $\idb$ are bound.