ODIn/paper-BagRelationalPDBsAreHard
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Boris Glavic 2021-04-08 21:51:36 -05:00
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ra-to-poly.tex
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@ -47,8 +47,7 @@ The reduced form of a lineage polynomial can be obtained but requires a linear s
\subsection{Probabilistic Databases (PDBs)}
An \textit{incomplete database} $\idb$ is a set of deterministic databases $\db$ called possible worlds.
Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\pdb$ is a pair $(\idb, \pd)$ where $\idb$ is an incomplete database and $\pd$ is a probability distribution over $\idb$. Queries over probabilistic databases are evaluated using the so-called possible world semantics. Under possible world semantics, the result of a query $\query$ over an incomplete database $\idb$ is the set of query answers produced by evaluating $\query$ over each possible world:
\[\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}\]
Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\pdb$ is a pair $(\idb, \pd)$ where $\idb$ is an incomplete database and $\pd$ is a probability distribution over $\idb$. Queries over probabilistic databases are evaluated using the so-called possible world semantics. Under possible world semantics, the result of a query $\query$ over an incomplete database $\idb$ is the set of query answers produced by evaluating $\query$ over each possible world: $\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}$
For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$ that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
\[\forall \db \in \query(\idb): \pd'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \pd(\db') \]
@ -105,7 +104,7 @@ $\semNX$-PDBs and a function $\rmod$ (which transforms an $\semNX$-PDB to an equ
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\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
Given an $\semN$-PDB $\pdb = (\idb,\pd)$ and $\semNX$-PDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, we have:
\[ \expct_{\idb \sim \pd}[\query(\idb)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\polyForTuple(\vct{W})}. \]
$ \expct_{\idb \sim \pd}[\query(\idb)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\polyForTuple(\vct{W})}. $
\end{Proposition}
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.
This proposition shows that computing expected tuple multiplicities is equivalent to computing the expectation of a polynomial (for that tuple) from a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$.
@ -122,3 +121,7 @@ A \emph{\ti} is a \bi where each block contains exactly one tuple.
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