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moving 1.6 and poly(*) to appendix

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Oliver Kennedy 2022-05-19 14:45:58 -04:00
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37
app_notation-background.tex
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@ -119,6 +119,28 @@ By expanding $\nxpolyqdt$ and the expectation we have:
\qed
\end{proof}
\subsection{Construction of the lineage (polynomial) for an $\raPlus$ query $\query$ over $\gentupset'$}
\label{fig:lin-poly-bidb}
The following rules are analogous to \Cref{fig:nxDBSemantics}, differing only in the base case. They are presented here for completeness.
\begin{align*}
\poly'\pbox{\project_A\inparen{\query}, \gentupset', \tup_j} =
&~\sum_{\substack{\tup_{j'},\\\project_{A}\inparen{\tup_{j'}} = \tup_j}}\poly'\pbox{\query, \gentupset', \tup_{j'}}\\
%
\poly'\pbox{\query_1\union\query_2, \gentupset', \tup_j} =
&\qquad\poly'\pbox{\query_1, \gentupset', \tup_j}+\poly'\pbox{\query_2, \gentupset', \tup_j}\\
%
\poly'\pbox{\select_\theta\inparen{\query}, \gentupset', \tup_j} =
&~\begin{cases}\theta = 1 &\poly'\pbox{\query, \gentupset', \tup_j}\\\theta = 0& 0\\\end{cases} \\
%
\poly'\pbox{\query_1\join\query_2, \gentupset', \tup_j} =
&\qquad \poly'\pbox{\query_1, \gentupset', \project_{attr\inparen{\query_1}}\inparen{\tup_j}}\\ &\qquad\cdot\poly'\pbox{\query_2, \gentupset', \project_{attr\inparen{\query_2}} \inparen{\tup_j}}\\
%
\poly'\pbox{\rel,\gentupset', \tup_j} =& j\cdot X_{\tup, j}.
\end{align*}\\%[-10mm]
\subsection{Proposition~\ref{proposition:q-qtilde}}\label{app:subsec-prop-q-qtilde}
\noindent Note the following fact:
@ -177,6 +199,21 @@ Note that~\Cref{lem:tidb-reduce-poly} shows that $\expct\pbox{\poly} =$ $\rpoly(
\qed
\end{proof}
\subsection{Definition of $\polyf(\cdot)$}
\label{def:poly-func}
We formally define the relationship of circuits with polynomials. While the definition assumes one sink for notational convenience, it easily generalizes to the multiple sinks case.
$\polyf(\circuit)$ maps the sink of circuit $\circuit$ to its corresponding polynomial in \abbrSMB. $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
\begin{equation*}
\polyf(\circuit) = \begin{cases}
\polyf(\circuit_\lchild) + \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circplus\\
\polyf(\circuit_\lchild) \cdot \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circmult\\
\circuit.\val &\text{ if \circuit.\type } = \var \text{ OR } \tnum.
\end{cases}
\end{equation*}
$\circuit$ need not encode $\poly\inparen{\vct{X}}$ in the same, default \abbrSMB representation. For instance, $\circuit$ could encode the factorized representation $(X + 2Y)(2X - Y)$ of $\poly\inparen{\vct{X}} = 2X^2+3XY-2Y^2$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = \poly\inparen{\vct{X}}$ is always the equivalent \abbrSMB representation.
%%% Local Variables:
%%% mode: latex

35
binarybidb.tex
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@ -49,7 +49,9 @@ Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in
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 $\worldvec$.}
\end{Definition}
\Cref{fig:lin-poly-bidb} shows the lineage construction of $\poly'\inparen{\vct{X}}$ given $\raPlus$ query $\query$ for arbitrary deterministic $\gentupset'$. Note that the semantics differ from~\Cref{fig:nxDBSemantics} only in the base case.
Lineage polynomials for arbitrary deterministic $\gentupset'$ are constructed in a manner analogous to $1$-\abbrTIDB\xplural (see \Cref{fig:nxDBSemantics}), differing only in the base case.
In a $1$-\abbrTIDB, each tuple contributes a multiplicity of 0 or 1, and $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$.
In a $c$-\abbrTIDB, each expanded tuple $t_j$ contributes its corresponding multiplicity: $\polyqdt{\rel}{\gentupset}{\tup_j} = j\cdot X_\tup$. These semantics are fully detailed in \Cref{fig:lin-poly-bidb}.
\begin{Proposition}[\abbrCTIDB reduction]\label{prop:ctidb-reduct}
Given \abbrCTIDB $\pdb =$\newline $\inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$ be the \emph{\abbrOneBIDB} obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intuple{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$ where $\bound_{\tup_j} = j$ for each $\tup_j$ in $\tupset'$.
@ -58,37 +60,6 @@ Given \abbrCTIDB $\pdb =$\newline $\inparen{\worlds, \bpd}$, let $\pdb' = \inpar
\end{Proposition}
We now define the reduced polynomial $\rpoly'$ of a \abbrOneBIDB.
\begin{figure}[t!]
%\centering
%\resizebox{0.5\textwidth}{!}{
%\begin{minipage}{0.5\textwidth}
\begin{align*}
&\begin{aligned}[t]
&\poly'\pbox{\project_A\inparen{\query}, \gentupset', \tup_j} =\\
&~\sum_{\substack{\tup_{j'},\\\project_{A}\inparen{\tup_{j'}} = \tup_j}}\poly'\pbox{\query, \gentupset', \tup_{j'}}
\end{aligned}
&
&\begin{aligned}[t]
&\poly'\pbox{\query_1\union\query_2, \gentupset', \tup_j} = \\
&\qquad\poly'\pbox{\query_1, \gentupset', \tup_j}+\poly'\pbox{\query_2, \gentupset', \tup_j}
\end{aligned}\\
&\begin{aligned}
&\poly'\pbox{\select_\theta\inparen{\query}, \gentupset', \tup_j} =\\
&~\begin{cases}\theta = 1 &\poly'\pbox{\query, \gentupset', \tup_j}\\\theta = 0& 0\\\end{cases}
\end{aligned}
&
&\begin{aligned}
&\poly'\pbox{\query_1\join\query_2, \gentupset', \tup_j} = \\
&\qquad \poly'\pbox{\query_1, \gentupset', \project_{attr\inparen{\query_1}}\inparen{\tup_j}}\\ &\qquad\cdot\poly'\pbox{\query_2, \gentupset', \project_{attr\inparen{\query_2}} \inparen{\tup_j}}
\end{aligned}\\
&&&\poly'\pbox{\rel,\gentupset', \tup_j} = j\cdot X_{\tup, j}.
\end{align*}\\%[-10mm]
%\end{minipage}}
\setlength{\abovecaptionskip}{-0.2cm}
\caption{Construction of the lineage (polynomial) for an $\raPlus$ query $\query$ over $\gentupset'$.}
\label{fig:lin-poly-bidb}
\vspace{-0.53cm}
\end{figure}
\begin{Definition}[$\rpoly'$]\label{def:reduced-poly-one-bidb}
Given a polynomial $\poly'\inparen{\vct{X}}$ generated from a \abbrOneBIDB, let $\rpoly'\inparen{\vct{X}}$ denote the reduced $\poly'\inparen{\vct{X}}$ derived as follows: i) compute $\smbOf{\poly'\inparen{\vct{X}}}$ eliminating monomials with cross terms $X_{\tup}X_{\tup'}$ for $\tup\neq \tup' \in \block_i$ and ii) reduce all \emph{variable} exponents $e > 1$ to $1$.

14
prob-def.tex
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@ -56,19 +56,9 @@ Note that the ciricuit \circuit representing $AX$ and the circuit \circuit' repr
\label{fig:circuit}
\vspace{-0.53cm}
\end{figure}
We next formally define the relationship of circuits with polynomials. While the definition assumes one sink for notational convenience, it easily generalizes to the multiple sinks case.
\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
$\polyf(\circuit)$ maps the sink of circuit $\circuit$ to its corresponding polynomial in \abbrSMB. $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
\begin{equation*}
\polyf(\circuit) = \begin{cases}
\polyf(\circuit_\lchild) + \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circplus\\
\polyf(\circuit_\lchild) \cdot \polyf(\circuit_\rchild) &\text{ if \circuit.\type } = \circmult\\
\circuit.\val &\text{ if \circuit.\type } = \var \text{ OR } \tnum.
\end{cases}
\end{equation*}
\end{Definition}
$\circuit$ need not encode $\poly\inparen{\vct{X}}$ in the same, default \abbrSMB representation. For instance, $\circuit$ could encode the factorized representation $(X + 2Y)(2X - Y)$ of $\poly\inparen{\vct{X}} = 2X^2+3XY-2Y^2$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = \poly\inparen{\vct{X}}$ is always the equivalent \abbrSMB representation.
We next define the inverse of the function $\polyf(\cdot)$ (\Cref{def:poly-func}), which maps a circuit to the polynomial it encodes.
\begin{Definition}[Circuit Set]\label{def:circuit-set}
$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.