Latest Version

poly-form.tex

@@ -64,6 +64,7 @@ We would like to argue that in the general case there is no computation of expec
 
 To this end, consider the following graph $G(V, E)$, where $|E| = m$, $|V| = \numTup$, and $i, j \in [\numTup]$.  Consider the query $q_E(X_1,\ldots, X_\numTup) = \sum\limits_{(i, j) \in E} X_i \cdot X_j$.
 \AR{The two lemmas need to be re-written once notation for representing a query is finalized in Section 1.}
+\AH{\^-----This is an issue that we are currently discussing.}
 \begin{Lemma}\label{lem:const-p}
 If we can compute $\poly(\wElem_1,\ldots, \wElem_\numTup) = q_E(\wElem_1,\ldots, \wElem_\numTup)^3$ in T(m) time for $\wElem_1 = \ldots = \wElem_\numTup = \prob$, then we can count the number of 3-matchings in $G$ in $T(m) + O(m)$ time.
 \end{Lemma}
@@ -164,3 +165,5 @@ Then $\numocc{\tri}_2 = 0$, and if we can prove that\AR{Again you are not transc
 \end{itemize}
 we solve our problem for $q_E^3$ based on $G_2$ and we can compute $\numocc{\threedis}$, a hard problem.
 \end{proof}
+{\bf TESTING}
+$\vec{w}\sim\mathcal{D}$

ra-to-poly.tex

@@ -1,11 +1,11 @@
 %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
-}
+%\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 $\poly(X_1,\ldots, X_\numTup)$.  To be clear,  $\poly(X_1,\ldots, X_\numTup)$ is a polynomial whose variables represent the tuple annotations of an arbitrary query.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.