From c50da5ab077f327959aa6a4846ae728ce33dd0e8 Mon Sep 17 00:00:00 2001 From: Boris Glavic Date: Tue, 15 Dec 2020 11:02:22 -0600 Subject: [PATCH] poly --- poly-form.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/poly-form.tex b/poly-form.tex index 669ed78..9b32caa 100644 --- a/poly-form.tex +++ b/poly-form.tex @@ -49,10 +49,10 @@ The degree of the running example polynomial is $2$. In this paper we consider o We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (\emph{\ti-lineage polynomial}), if there exists an n-ary $\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$), and n-ary tuple $\tup$ such that $\query(\vct{X}) = \query(\pxdb)(\tup)$. % Before proceeding, note that the following is assume that polynomials are \bis (which subsume \tis as a special case). Note the \tis are a special case of \bis and, thus, the following applies to \tis as well. -Recall that in a \bi $\pdbx$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pdbx$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\vct{p}(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\vct{p}$ and the blocks induce a the probability distribution $\pd$ of $\pdbx$. +Recall that in a \bi $\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\vct{p}(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\vct{p}$ and the blocks induce a the probability distribution $\pd$ of $\pxdb$. We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$. -% and the probability distribution of $\pdbx$ is uniquely determined based on a probability vector $\vct{p}$ that associates each tuple a probability +% and the probability distribution of $\pxdb$ is uniquely determined based on a probability vector $\vct{p}$ that associates each tuple a probability % variables are independent of each other (or disjoint if they are from the same block) and each variable $X$ is associated with a probability $\vct{p}(X) = \pd[X = 1]$. Thus, we are dealing with polynomials $\poly(\vct{X})$ that are annotations of a tuple in the result of a query $\query$ over a BIDB $\pxdb$ where $\vct{X}$ is the set of variables that occur in annotations of tuples of $\pxdb$. % While the definition of polynomial $\poly(\vct{X})$ over a $\bi$ input doesn't change, we introduce an alternative notation which will come in handy. Given $\ell$ blocks, we write $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.