Update on Overleaf.

ra-to-poly.tex

@@ -57,13 +57,13 @@ or simply lineage polynomial), if it is clear from context that there exists an
 %A block independent database \abbrBIDB $\pdb'$ can viewed as a $1$-\abbrTIDB $\pdb$ with the added flexibility that each $\tup\in\tupset$ has multiple disjoint alternatives, i.e., all $\tup \in \tupset'$ are partitioned into $m$ independent blocks with the condition that tuples $\tup \in \block_i$ for $i \in \pbox{m}$ are disjoint events.  We define next a specific construction of \abbrBIDB that is useful for our work.
 %}
 \begin{Definition}[\abbrOneBIDB]\label{def:one-bidb}
-Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$  where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$.  $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples.  $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup \in \block_i}\prob_\tup \leq 1$.  Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_i = \begin{cases}
+Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$  where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$.  $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples.  $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup \in \block_i}\prob_\tup \leq 1$.  Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_i(W) = \begin{cases}
 	1 - \sum_{\tup\in\block_i}\prob_\tup	&	\text{if }W_\tup = 0\text{ for all }\tup\in\block_i\\
-	0									&	\text{if there exists } \tup,~\tup'\in\block_i, W_\tup, W_{\tup'}\geq 1\\
-	\prob_\tup							&	W_\tup = 1.\\
+	0									&	\text{if there exists } \tup,~\tup'\in\block_i, W_\tup, W_{\tup'}\neq 0\\
+	\prob_\tup							&	W_\tup \ne 0 \text{ for the unique } t\in B_i.\\
 	\end{cases}$
 	
-\noindent$\bpd'$ is the probability distribution across all worlds such that, given $W\in\bigtimes_{\tup \in \tupset'}\inset{0,\bound_\tup}$, $\probOf\pbox{\worldvec = W} = \prod_{i\in\pbox{\numblock}}\prob_{i}$.
+\noindent$\bpd'$ is the probability distribution across all worlds such that, given $W\in\bigtimes_{\tup \in \tupset'}\inset{0,\bound_\tup}$, $\probOf\pbox{\worldvec = W} = \prod_{i\in\pbox{\numblock}}\prob_{i}(W)$.
 % if for any $i \in\pbox{\numblock}$ there does \emph{not} exist a $\tup\neq\tup' \in \block_i$ such that $W_{\tup}, W_{\tup'} \geq 1$, where $\prob_{\tup}$ is the marginal probability $\tup$.  Otherwise, $\probOf\pbox{\worldvec=W} = 0$.\
 \footnote{
 We slightly abuse notation here, denoting a world vector as $W$ rather than $\worldvec$ to distinguish between the random variable and the world instance.  When there is no ambiguity, we will denote a world vector as $\worldvec$.}% $\worldvec\in\prod_{\tup\in\tupset'}\inset{0,\bound_\tup},\tup,~\tup'\in\block_i~:~\probOf\pbox{\worldvec_\tup, \worldvec_\tup'>0} = 0$.
@@ -73,7 +73,9 @@ We now present a reduction that is useful in deriving our results:
 
 \begin{Proposition}[\abbrCTIDB reduction]\label{def:ctidb-reduct}
 Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$ be the \emph{\abbrOneBIDB} obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$.% such that $X_{\tup, j}\in\inset{0,1}$.
-  The probability distribution $\bpd'$ is the characterized by the vector $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ for $\tup\in\tupset$ with multiplicity $j$.%and the \abbrBIDB disjoint requirement, where given any $\worldvec\in\onebidbworlds{\tupset'}$, $\probOf\pbox{\worldvec_{\tup, j}, \worldvec_{\tup, j'} > 0} = 0$ for any $j \neq j' \in \pbox{\bound}$.%, such that for any $W\in\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}$, $\probOf\pbox{\worldvec = W} = \prod_{\tup\in\tupset', j\in\pbox{\bound}}W_{\tup, j}\cdot j\cdot\prob_\tup$ if $\forall \tup \in \tupset'\not\exists j\neq j'\in\pbox{\bound}, W_{\tup, j}, W_{\tup, j'} \geq 1$; otherwise $\probOf\pbox{\worldvec = W} = 0$.% that for any $X_{\tup, j} = 1, j'\in\pbox{\bound} - \inset{j}, X_{\tup, j'} = 0$.
+  The probability distribution $\bpd'$ is the characterized by the vector $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$. % for $\tup\in\tupset$ with multiplicity $j$.
+  Then, the distributions $\mathcal{P}$ and $\mathcal{P}'$ are equivalent.
+  %and the \abbrBIDB disjoint requirement, where given any $\worldvec\in\onebidbworlds{\tupset'}$, $\probOf\pbox{\worldvec_{\tup, j}, \worldvec_{\tup, j'} > 0} = 0$ for any $j \neq j' \in \pbox{\bound}$.%, such that for any $W\in\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}$, $\probOf\pbox{\worldvec = W} = \prod_{\tup\in\tupset', j\in\pbox{\bound}}W_{\tup, j}\cdot j\cdot\prob_\tup$ if $\forall \tup \in \tupset'\not\exists j\neq j'\in\pbox{\bound}, W_{\tup, j}, W_{\tup, j'} \geq 1$; otherwise $\probOf\pbox{\worldvec = W} = 0$.% that for any $X_{\tup, j} = 1, j'\in\pbox{\bound} - \inset{j}, X_{\tup, j'} = 0$.
 % $\block_\tup,~j\in\pbox{\bound}~|~X_{\tup, j} = 1,\not\exists j'\neq j~|~X_{\tup, j'} = 1$. 
 %$\tup_j\geq1\implies \tup_{j'} = 0$.$\forall j, j' \in \pbox{\bound},\forall \tup\in\tupset, \tup_j\geq 1\implies \tup_{j'} = 0$ for any block $\block_\tup$. 
 \end{Proposition}