paper-BagRelationalPDBsAreHard/ra-to-poly.tex

%root: main.tex

\section{Query translation into polynomials}
\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.}