Changes to S 2

ra-to-poly.tex

@@ -50,18 +50,22 @@ We focus on the problem of computing $\expct_\pdassign\pbox{\apolyqdt\inparen{\r
 \label{subsec:tidbs-and-bidbs}
 In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
 %
-A \bi $\pdb$ is a \abbrPDB  such that (i) every tuple is annotated with either $0$ (i.e., the tuple does not exist) or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pdb$ for which $\pdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same block are disjoint events.
-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 \tis and \bis in greater detail.
-%
-In a \bi (and by extension a \ti), tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
-  Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function).  For $t_{i, j} \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
-}
-Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ and the blocks induce the probability distribution $\pd$ of $\pdb$.
-We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
-$\poly(\vct{X})$ = $\poly(X_{1, 1},\ldots, X_{1, \abs{\block_1}},$ $\ldots, X_{\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$.\footnote{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 $n=\sum_{i=1}^\ell \abs{b_i}$.}
+A \bi $\pdb$ is a \abbrPDB with the constraint that 
+%(i) every tuple $\tup_i$ is annotated with a unique random variable $\randWorld_i \in \{0, 1\}$ and (ii) that 
+the tuples can be partitioned into a set of $\ell$ blocks such that tuples $\tup_{i, j}, \tup_{k, j'}$ from separate blocks $(i\neq k, j \in [\abs{i}], j' \in [\abs{k}])$ are independent of each other while tuples $\tup_{i, j}, \tup_{i, k}$ from the same block are disjoint events.\footnote{
+  Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function).  For $t_{i, j} \in b_i$, the event $(\randWorld_{i,j} = 1)$ corresponds to the event $(\randWorld_i = j)$ in the customary annotation scheme.
+}  
+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 $\pdb$ is the one induced from individual tuple probabilities $\prob_{i, j}$ and the conditions on the blocks.  A \abbrTIDB is a \abbrBIDB with the added requirement that each block is size $1$.
+
+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 $k = \sum_i \abs{b_i} + j$).  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 $\randWorld_{i, j} = \randWorld_{i, k} = 1$, 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.
+%%
+%In a \bi (and by extension a \ti), tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.
+%Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ and the blocks induce the probability distribution $\pd$ of $\pdb$.
+%We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
+%$\poly(\vct{X})$ = $\poly(X_{1, 1},\ldots, X_{1, \abs{\block_1}},$ $\ldots, X_{\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$.
 
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