Tidying up.

ra-to-poly.tex

@@ -12,23 +12,6 @@ A \textit{probabilistic database} $\pdb$ is a pair $(\idb, \pd)$ where $\idb$ is
 
 For a probabilistic  database $\pdb = (\idb, \pd)$,  the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$  that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
 
-%Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
-%We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}.
-% $\semNX$-databases are functions from tuples to elements of $\semNX$, typically called annotations.
-%Given an $\semNX$-database $\db$,  it is common to use $\db(\tup)$ to denote the polynomial annotating tuple $\tup$ in $\db$.
-%%Note that based on this definition of $\rel$, $\rel(\tup)$ is the lineage polynomial for $\tup$.
-%Let $\numvar$ be the number of tuples in $\pdb$.  Then, each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
-%The multiplicity of $\tup \in \db$, denoted $\db(\tup)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $\tup$ on $\vct{\wElem}$.
-%$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
-%
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-%
-%We will use $\semNX$-\abbrPDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$ over the assignments to $\vct{X}$.
-%We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-\abbrPDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{\numvar} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
-%$\semNX$-\abbrPDB\xplural and a function $\rmod$ (which transforms an $\semNX$-\abbrPDB to  a classical bag-\abbrPDB, or $\semN$-\abbrPDB~\cite{DBLP:conf/pods/GreenKT07,feng:2019:sigmod:uncertainty}) are both formalized in \Cref{subsec:supp-mat-background}.
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-%END: move to appendix.
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 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}
@@ -51,8 +34,6 @@ the tuples in $\dbbase$ can be partitioned into a set of $\ell$ blocks such that
 Each tuple $\tup_{i, j}$ is annotated with a random variable $\randWorld_{i, j} \in \{0, 1\}$ denoting its presence in a possible world $\db$.  The probability distribution $\pd$ over $\dbbase$ is the one induced from individual tuple probabilities $\prob_{i, j}\in \vct{\prob}=\inparen{\prob_{1, 1},\ldots,\prob_{\abs{\block},\ldots,\abs{\block_{\abs{\block}}}}}$ and the conditions on the blocks.  A \abbrTIDB is a \abbrBIDB where each block has size exactly $1$.
 
 Instead of looking only at the possible worlds of $\pdb$, one can consider all worlds, including those that cannot exist due to disjointness.  The all worlds set can be modeled by $\vct{\randWorld}\in \{0, 1\}^\numvar$,\footnote{Here and later on in the paper, especially in \Cref{sec:algo}, we will overload notation and rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$.} such that $\randWorld_k \in \vct{\randWorld}$ represents the presence of $\tup_{i, j}$ (where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).  We denote a probability distribution over all $\vct{\randWorld} \in \{0, 1\}^\numvar$ as $\pdassign$.  When $\pdassign$ is the one induced from each $\prob_{i, j}$ while assigning $\probOf\pbox{\vct{\randWorld}} = 0$ for any $\vct{\randWorld}$ with $\randWorld_{i, j} = \randWorld_{i, k} = 1$ for any block $i$ and $j\neq k$, we end up with a bijective mapping from $\pd$ to $\pdassign$, such that each mapping is equivalent, implying the distributions are equivalent.
-%that $\forall i \in \abs{\block}, \forall j\neq k \in [\block_i] \suchthat \db\inparen{\tup_{i, j}} = 0 \vee \db\inparen{\tup_{i, k} = 0}$.In other words, each random variable corresponds to the event of a single tuple's presence.
-%A \emph{\ti} is a \bi where each block contains exactly one tuple.
 \Cref{subsec:supp-mat-ti-bi-def} explains \abbrTIDB\xplural and \abbrBIDB\xplural in greater detail.