ODIn/paper-BagRelationalPDBsAreHard
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Aaron Huber 2020-12-15 18:46:36 -05:00
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ra-to-poly.tex
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@ -21,7 +21,7 @@ Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \t
A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ ($\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to tuple $t$.
We review the semantics of positive relational algebra queries over $\semK$-relations below.
Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural number. $\semN$-databases are used to model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-databases ($\semN$-PDB) is a PDB where each possible world is a $\semN$-database. We will study the problem of evaluating statical moments of query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result tuple $t$, we treat $\query(\db)(t)$ as a random $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \pd}[\query(\db)(t)]$:
Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases are used to model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-databases ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We will study the problem of evaluating statistical moments of query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result tuple $t$, we treat $\query(\db)(t)$ as a random $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \pd}[\query(\db)(t)]$:
\begin{align}\label{eq:bag-expectation}
\expct_{\idb \sim \pd}[\query(\db)(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \pd(\db)
@ -36,7 +36,7 @@ For completeness, we briefly review the semantics for $\raPlus$ queries over $\s
We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$. In the definition shown below, we assume that tuples are of appropriate arity and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$. Furthermore, $\theta(\tup)$ denotes the (boolean) result of evaluating condition $\theta$ over $\tup$.
\begin{align*}
& \evald{\project_A(\rel)}{\db}(\tup) & & = & & \sum_{\tup': \project_A(\tup) = \tup} \evald{\rel}{\db}(\tup') \\
& \evald{\project_A(\rel)}{\db}(\tup) & & = & & \sum_{\tup': \project_A(\tup') = \tup} \evald{\rel}{\db}(\tup') \\
& \evald{(\rel_1 \union \rel_2)}{\db}(\tup) & & = & & \evald{\rel_1}{\db}(\tup) \addK \evald{\rel_2}{\db}(\tup) \\
& \evald{(\rel_1 \join \rel_2)}{\db}(\tup) & & = & & \evald{\rel_1}{\db}(\project_{\sch(\rel_1)}(\tup)) \multK \evald{\rel_2}{\db}(\project_{\sch(\rel_2)}(\tup)) \\
& \evald{\select_\theta(\rel)}{\db}(\tup) & & = & & \begin{cases}
@ -54,7 +54,7 @@ Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$ an
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[Representation System]\label{def:representation-syste}
A representation system for $\semN$-PDBs is a tuple $(\reprs, \rmod)$ where $\reprs$ is a set of representations and $\rmod$ associates which each $\repr \in \reprs$ a $\semN$-PDB $\pdb$. We say that a representation system is \emph{closed} under a class of queries $\qClass$ if for any query $\query \in \qClass$ we have:
A representation system for $\semN$-PDBs is a tuple $(\reprs, \rmod)$ where $\reprs$ is a set of representations and $\rmod$ associates with each $\repr \in \reprs$ an $\semN$-PDB $\pdb$. We say that a representation system is \emph{closed} under a class of queries $\qClass$ if for any query $\query \in \qClass$ we have:
%
\[ \rmod(\query(\repr)) = \query(\rmod(\repr)) \]