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Boris Glavic 2019-12-01 15:58:38 -06:00
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Documents/foundations.tex
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@ -1019,16 +1019,16 @@ where the ${\cal A}^i$ are relational database instances (possible worlds)
or, as a commutative diagram,
\vspace*{1em}
\begin{center}
\begin{psmatrix}[colsep=8em,rowsep=5em,nodesepA=3pt,nodesepB=3pt]
$T$ & $\overline{Q}(T)$\\
$\{{\cal A}^1,\ldots,{\cal A}^n\}$ &
$\{Q({\cal A}^1),\ldots,Q({\cal A}^n)\}$
%
\ncline{->}{1,1}{2,1}<{\textit{rep}}
\ncline{->}{2,1}{2,2}^{$Q$}
\ncline{->}{1,1}{1,2}^{$\overline{Q}$}
\ncline{->}{1,2}{2,2}>{\textit{rep}}
\end{psmatrix}
% \begin{psmatrix}[colsep=8em,rowsep=5em,nodesepA=3pt,nodesepB=3pt]
% $T$ & $\overline{Q}(T)$\\
% $\{{\cal A}^1,\ldots,{\cal A}^n\}$ &
% $\{Q({\cal A}^1),\ldots,Q({\cal A}^n)\}$
% %
% \ncline{->}{1,1}{2,1}<{\textit{rep}}
% \ncline{->}{2,1}{2,2}^{$Q$}
% \ncline{->}{1,1}{1,2}^{$\overline{Q}$}
% \ncline{->}{1,2}{2,2}>{\textit{rep}}
% \end{psmatrix}
\end{center}
@ -1351,11 +1351,3 @@ Full probabilistic world-set algebra is essentially not harder than the language
It is worth noting that repair-key by itself, despite the blowup of possible worlds, does not make queries hard. For the language consisting of positive relational algebra, repair-key, and poss, we have shown by construction that it has PTIME complexity: We have given a positive relational algebra rewriting to queries on the representations earlier in this section. Thus queries are even in the highly parallelizable complexity class AC$_0$.
The final result in Figure~\ref{tab:complexity} concerns the language consisting of the positive relational algebra operations, repair-key, $(\epsilon, \delta)$-approximation of confidence computation, and the generalized equality generating dependencies of \cite{Koch2008} for which we can rewrite difference of uncertain relations to difference of confidence values (see Example~\ref{ex:trick}). The result is that queries of that language that close the possible worlds semantics -- i.e., that use conf to compute a certain relation -- are in PTIME overall.