From 7cc07ae2b7f34d1286df829893acd027d29ad3d0 Mon Sep 17 00:00:00 2001 From: Atri Rudra Date: Fri, 10 Jun 2022 21:05:14 +0000 Subject: [PATCH] Update on Overleaf. --- introduction.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/introduction.tex b/introduction.tex index 3a85c64..9c66b38 100644 --- a/introduction.tex +++ b/introduction.tex @@ -7,7 +7,7 @@ Formally, a \abbrCTIDB, $\pdb = \inparen{\worlds, \bpd}$ is defined over a \dbbaseName (i.e. a `base' set of tuples) $\tupset$ and a probability distribution $\bpd$ over all possible worlds generated by assigning each tuple $\tup \in \tupset$ a multiplicity in the range $[0,\bound]$. Any such world can be encoded as a vector (of length $\numvar=\abs{\tupset}$) from $\worlds$, such that the multiplicity of each $\tup \in \tupset$ is stored at a distinct index. A given world $\worldvec \in\worlds$ can be interpreted as follows: for each $\tup \in \tupset$, $\worldvec_{\tup}$ is the multiplicity of $\tup$ in $\worldvec$. -We note that encoding a possible world as a vector, while non-standard, is equivalent to encoding it as a bag of tuples (\Cref{prop:expection-of-polynom}). +We note that encoding a possible world as a vector, while non-standard, is equivalent to encoding it as a bag of tuples (\Cref{app:subsec:background-nxdbs}). %(\Cref{prop:expection-of-polynom}). %in \Cref{subsec:expectation-of-polynom-proof}). Given that tuple multiplicities are independent events, the probability distribution $\bpd$ can be expressed compactly by assigning each tuple a probability distribution over $[0,\bound]$. Let $\prob_{\tup,j}$ denote the probability that tuple $\tup$ is assigned multiplicity $j$. The probability of a world $\worldvec$ is then $\prod_{\tup \in \tupset} \prob_{\tup,j(t)}$ for $j(t) = \worldvec_{\tup}$. %