ODIn/paper-BagRelationalPDBsAreHard
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Started translation, notation section

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Aaron Huber 2020-06-23 15:49:19 -04:00
parent 29afac852e
commit 33fc3d74d6
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20
macros.tex
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@ -3,6 +3,17 @@
% NOTATION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%RA-to-Poly Notation
\newcommand{\rel}{R}
\newcommand{\reli}{S}
\newcommand{\relii}{T}
\newcommand{\query}{Q}
\newcommand{\join}{\Join}
\newcommand{\select}{\sigma}
\newcommand{\project}{\pi}
\newcommand{\union}{\cup}
%Polynomial Reformulation
\newcommand{\wbit}{w}
\newcommand{\expct}{\mathop{\mathbb{E}}}
@ -324,8 +335,8 @@
\newcommand{\schemaOf}{\textsc{Sch}}
\newcommand{\arity}[1]{arity({#1})}
\newcommand{\db}{D}
\newcommand{\rel}{R}
\newcommand{\query}{Q}
%\newcommand{\rel}{R}
%\newcommand{\query}{Q}
\newcommand{\qClass}{\mathcal{C}}
% \newcommand{\uDom}{\mathcal{U}}
\newcommand{\tup}{t}
@ -339,10 +350,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Algebra
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\selection}{\sigma}
\newcommand{\projection}{\pi}
\newcommand{\join}{\Join}
\newcommand{\union}{\cup}
\newcommand{\intersection}{\cap}
\newcommand{\rename}{\rho}
\newcommand{\aggregation}[2]{{}_{#1}\gamma_{#2}}

25
ra-to-poly.tex
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@ -1,6 +1,29 @@
%root: main.tex
\section{Query translation into polynomials}
This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}.
\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}
1) RA notation
2) DB (TIDB) notation
3) How queries translate into polynomials
}
Given tables $\rel, \reli$, an arbitrary query $\query(\rel)$ over the positive relational operators (SPJU), abusing notation slightly denote the query polynomial as $\poly(X_1,\ldots, X_\numTup)$. The annotation for arbitrary tuple $\tup$ can be viewed as an element of the image of $\rel$, where relation $\rel$ can be thought of as a function with preimage of all tuples in $\rel$, such that $\rel(\tup) = \poly(X_1,\ldots, X_\numTup)$. Further, it is known that the algebraic semiring structure aptly models the translation and computation of query operations into tuple annotation, aka polynomials.
To make things more concrete, consider the $\{\mathbb{N}, \times, +, 1, 0\}$ bag semiring. Here the set in which the tuple annotations (computed polynomials) exist is the natural numbers. Query operations are translated into one of the two semiring operators, with $\project$ and $\union$ of agreeing tuples being the equivalent of the '+' opertator in polynomial $\poly$, $\join$ translating into the $\times$ operator, and finally, $\select$ is better modeled as a function that returns either $\rel(\tup)$ or $0$ based on some predicate.
Consider the translation of relational operators to polynomial operators in greater detail.
\begin{align*}
&\project_A(\rel)(\tup) = &&\sum_{\tup' s.t. \tup[A] = \tup'} \rel(\tup')\\
& (\rel_1 \union \rel_2)(\tup) = &&\rel_1(\tup) + \rel_2(\tup)\\
&(\rel_1 \join_\theta \rel_2)(\tup) = &&\begin{cases}
\rel_1(\tup_1) \times \rel_2(\tup_2) &\text{if }\theta(\tup_1, \tup_2)\\
0 &\text{otherwise}
\end{cases} \\
&\select_\theta(\rel) = &&\begin{cases}
\rel(\tup) &\text{if }\theta(\tup) = 1\\
0 &\text{otherwise}.
\end{cases}
\end{align*}
\AR{This section needs to be written up {\bf before} you go forwad in the next section. In particular, this section should setup definitions/notation (e.g. how you to denote a general query on TIDB) that will be used later on so better get thet setup first.}