Changed redux definition to be more specific on how P' is derived.

mult_distinct_p.tex

@@ -58,14 +58,15 @@ SELECT COUNT(*) FROM $R_1$ JOIN $R_2$ JOIN$\cdots$JOIN $R_k$
 \noindent Consider again the \abbrCTIDB instance $\pdb$ of~\Cref{fig:two-step} and, for our hard instance, let $\bound = 1$.  $\pdb$ generalizes to one compatible to~\Cref{def:qk} as follows. Relation $T$ has $n$ tuples corresponding to each vertex for $i$ in $[n]$, each with probability $\prob_i$ and $R$ has tuples corresponding to the edges $\edgeSet$ (each with probability of $1$).\footnote{Technically, $\poly_{G}^\kElem(\vct{X})$ should have variables corresponding to tuples in $R$ as well, but since they always are present with probability $1$, we drop those. Our argument also works when all the tuples in $R$ also are present with probability $\prob$ but to simplify notation we assign probability $1$ to edges.}
 In other words, for this instance $\tupset$ contains the set of $\numvar$ unary tuples in $T$ (which corresponds to $\vset$) and $\numedge$ binary tuples in $R$ (which corresponds to $\edgeSet$).
 Note that this implies that $\poly_{G}^\kElem$ is indeed a \abbrCTIDB-lineage polynomial. % for a \abbrTIDB \abbrPDB.
-
+\AH{
+\textbf{@atri}, we discussed this last meeting, but I am not sure if we really pinpointed how we want to treat (\emph{in a consistent manner}) the runtime of~\Cref{lem:tdet-om} since $k$ is a constant and $m$ is growing.  Would it be a good idea to be consistent with the $O_\epsilon$ notation of~\Cref{prob:big-o-joint-steps} and say $O_k(\numedge)$}
 Next, we note that the runtime for answering $\query^k$ on deterministic database $\tupset$, as defined above, is $\bigO{\numedge}$ (i.e. deterministic query processing is `easy' for this query):
 \begin{Lemma}\label{lem:tdet-om}
 Let $\query^k$ and $\tupset$ be as defined above. Then
 % of \Cref{def:qk}, the runtime 
 $\qruntimenoopt{\query^k, \tupset}$ is $\bigO{\kElem\numedge}$.
 \end{Lemma}
-\AH{Should the above be $\qruntimenoopt{}$ or $\qruntime{}$?}
+
 \subsection{Multiple Distinct $\prob$ Values}
 \label{sec:multiple-p}
 %Unless otherwise noted, all proofs for this section are in \Cref{app:single-mult-p}.

ra-to-poly.tex

@@ -60,7 +60,8 @@ We now present a reduction that is useful in deriving our results:
 
 \begin{Definition}[\abbrCTIDB reduction]\label{def:ctidb-reduct}
 Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}, \bpd'}$ be the \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 one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement, where given any $\worldvec\in\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}$, $\probOf\pbox{\worldvec_{\tup, j}, \worldvec_{\tup, j'} > 0} = 0$ for any $j \neq j' \in \pbox{\bound}$.% that for any $X_{\tup, j} = 1, j'\in\pbox{\bound} - \inset{j}, X_{\tup, j'} = 0$.
+The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement, where given any $\worldvec\in\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}$, $\probOf\pbox{\worldvec_{\tup, j}, \worldvec_{\tup, j'} > 0} = 0$ 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 $\probOf\pbox{\worldvec = W} = 0$.\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 $\worldvec$.}% 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{Definition}