From ccc2f2ef0b5cb962ae5c6b935aede15981b67742 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Tue, 15 Dec 2020 18:46:36 -0500 Subject: [PATCH] Minor cosmetic changes --- ra-to-poly.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/ra-to-poly.tex b/ra-to-poly.tex index 4b66e96..f10bc86 100644 --- a/ra-to-poly.tex +++ b/ra-to-poly.tex @@ -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: % \[ \rmod(\query(\repr)) = \query(\rmod(\repr)) \]