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Atri Rudra 2022-06-09 18:28:47 +00:00 committed by node
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introduction.tex
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@ -28,31 +28,31 @@ An $\raPlus$ query is a query expressed in positive relational algebra, i.e., us
\begin{figure}[t!]
\begin{align*}
&\begin{aligned}[t]
&\polyqdt{\project_A(\query)}{\gentupset}{\tup} =\\
&~~\sum_{\tup': \project_A(\tup') = \tup} \polyqdt{\query}{\gentupset}{\tup'}
&\polyqdt{\project_A(\query)}{\gentupset}{\tup}\inparen{\vct{X}} =\\
&~\sum_{\tup': \project_A(\tup') = \tup} \polyqdt{\query}{\gentupset}{\tup'}\inparen{\vct{X}}
\end{aligned}
&
&\begin{aligned}[t]
&\polyqdt{\query_1 \union \query_2}{\gentupset}{\tup} =\\
&\qquad \polyqdt{\query_1}{\gentupset}{\tup} + \polyqdt{\query_2}{\gentupset}{\tup}\\
&\polyqdt{\query_1 \union \query_2}{\gentupset}{\tup}\inparen{\vct{X}} =\\
&~ \polyqdt{\query_1}{\gentupset}{\tup}\inparen{\vct{X}} + \polyqdt{\query_2}{\gentupset}{\tup}\inparen{\vct{X}}\\
\end{aligned}\\
&\begin{aligned}
&\polyqdt{\select_\theta(\query)}{\gentupset}{\tup} =\\
&~~ \begin{cases}
\polyqdt{\query}{\gentupset}{\tup} & \text{if }\theta(\tup) \\
&\polyqdt{\select_\theta(\query)}{\gentupset}{\tup}\inparen{\vct{X}} =\\
&~ \begin{cases}
\polyqdt{\query}{\gentupset}{\tup}\inparen{\vct{X}} & \text{if }\theta(\tup) \\
0 & \text{otherwise}.
\end{cases}
\end{aligned}
&
&\begin{aligned}
&\polyqdt{\query_1 \join \query_2}{\gentupset}{\tup} =\\
&\qquad\polyqdt{\query_1}{\gentupset}{\project_{\attr{\query_1}}{\tup}}\\
&\qquad\cdot\polyqdt{\query_2}{\gentupset}{\project_{\attr{\query_2}}{\tup}}
&\polyqdt{\query_1 \join \query_2}{\gentupset}{\tup}\inparen{\vct{X}} =\\
&~~\polyqdt{\query_1}{\gentupset}{\project_{\attr{\query_1}}{\tup}}\inparen{\vct{X}}\\
&~\cdot\polyqdt{\query_2}{\gentupset}{\project_{\attr{\query_2}}{\tup}}\inparen{\vct{X}}
\end{aligned}
\end{align*}%\\[-10mm]
%\setlength{\abovecaptionskip}{-0.25cm}
\savecaptionspace{
\caption{Lineage polynomial semantics given $\raPlus$ query $\query$, arbitrary \dbbaseName $\gentupset$ with variables $\inparen{X_\tup}_{\tup \in\gentupset}$, where for $\rel\in\gentupset$, $\tup\in\rel$, the base case is $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$.}% for any $\rel\in\gentupset$ and $\tup\in\rel$.}% consists of all $X_\tup$ over all $\rel$ in $\gentupset$ and $\tup$ in $\rel$, such that the base case $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$.} %Here $\gentupset.\rel$ denotes the instance of relation $\rel$ in $\gentupset$. Please note, after we introduce the reduction to $1$-\abbrBIDB, the base case will be expressed alternatively. The base case is $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$}
\caption{Lineage polynomial semantics for any $\raPlus$ query $\query$, arbitrary \dbbaseName $\gentupset$ (with variables $\vct{X}=\inparen{X_\tup}_{\tup \in\gentupset}$), where for $\rel\in\gentupset$, $\tup\in\rel$, the base case is $\polyqdt{\rel}{\gentupset}{\tup}\inparen{\vct{X}} = X_\tup$.}% for any $\rel\in\gentupset$ and $\tup\in\rel$.}% consists of all $X_\tup$ over all $\rel$ in $\gentupset$ and $\tup$ in $\rel$, such that the base case $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$.} %Here $\gentupset.\rel$ denotes the instance of relation $\rel$ in $\gentupset$. Please note, after we introduce the reduction to $1$-\abbrBIDB, the base case will be expressed alternatively. The base case is $\polyqdt{\rel}{\gentupset}{\tup} = X_\tup$}
\label{fig:nxDBSemantics}
}{\abovecapshrink}{\belowcapshrink}
%\vspace{-0.53cm}