Update on Overleaf.

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Atri Rudra 2022-06-10 21:05:14 +00:00 committed by node
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@ -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 $j(t) = \worldvec_{\tup}$.
%