Merge branch 'master' of gitlab.odin.cse.buffalo.edu:ahuber/SketchingWorlds
This commit is contained in:
Boris Glavic
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1cac699484
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%root: main.tex
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\section{$1 \pm \epsilon$ Approximation Algorithm}
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Since it is the case that computing the expected multiplicity of a compressed representation of a bag polynomial is hard, it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.
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\begin{Definition}[$\bi$~\cite{DBLP:series/synthesis/2011Suciu}]
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A Block Independent Database ($\bi$) is a PDB whose tuples are partitioned in blocks, where we denote block $i$ as $\block_i$. Each $\block_i$ is independent of all other blocks, while all tuples sharing the same $\block_i$ are mutually exclusive.
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\end{Definition}
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A $\ti$ is also a $\bi$ where each tuple is its own block.
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While the definition of polynomial $\poly(\vct{X})$ over a $\bi$ input doesn't change, we introduce an alternative notation which will come in handy. Given $\ell$ blocks, we write $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
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The number of tuples in the $\bi$ instance can be computed as $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .
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When considering $\bi$ input, it becomes necessary to redefine $\rpoly(\vct{X})$.
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\begin{Definition}[$\rpoly$ $\bi$ Redefinition]
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A polynomial $\poly(\vct{X})$ over a $\bi$ instance is reduced to $\rpoly(\vct{X})$ with the following criteria. First, all exponents $e > 1$ are reduced to $e = 1$. Second, all monomials sharing the same $\block$ are dropped. Formally this is expressed as
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\begin{equation*}
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\rpoly(\vct{X}) = \poly(\vct{X}) \mod X_i^2 - X_i \mod X_{\block_s, t}X_{\block_s, u}
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\end{equation*}
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for all $i$ in $[\numvar]$ and for all $s$ in $\ell$, such that for all $t, u$ in $[\abs{block_s}]$, $t \neq u$.
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\end{Definition}
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We state the approximation algorithm in terms of a $\bi$.
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First, let us introduce some useful definitions and notation. For illustrative purposes in the definitions below, let us consider when $\poly(\vct{X}) = 2x^2 + 3xy - 2y^2$.
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\begin{Definition}[Degree]\label{def:degree}
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The degree of polynomial $\poly(\vct{X})$ is the maximum sum of the exponents of a monomial, over all monomials.
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\end{Definition}
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The degree of $\poly(\vct{X})$ in the above example is $2$. In this paper we consider only finite degree polynomials.
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\begin{Definition}[Expression Tree]\label{def:express-tree}
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An expression tree $\etree$ is a binary %an ADT logically viewed as an n-ary
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tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either from the set $\mathbb{R}$ $(\tnum)$ or from the set of monomials $(\var)$. The members of $\etree$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\etree$ (i.e. one of $\{+, \times, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of $\etree$'s children where $\etree_\lchild$ is the left child and $\etree_\rchild$ the right child. Remaining fields hold values whose semantics we will fix later. When $\etree$ is used as input of ~\cref{alg:mon-sam} and ~\cref{alg:one-pass}, the values of \vari{partial} and \vari{weight} will not be set. %SEMANTICS FOR \etree: \vari{partial} is the sum of $\etree$'s coefficients , n, and \vari{weight} is the probability of $\etree$ being sampled.
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\end{Definition}
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Note that $\etree$ need not encode an expression in the standard monomial basis. For example, instead of our running example, $\etree$ could represent a compressed form such as $(x + 2y)(2x - y)$.
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Note that $\etree$ need not encode an expression in the standard monomial basis. For instance, $\etree$ could represent a compressed form of the running example, such as $(x + 2y)(2x - y)$.
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\begin{Definition}[poly$(\cdot)$]\label{def:poly-func}
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Denote $poly(\etree)$ to be the function that takes as input expression tree $\etree$ and outputs its corresponding polynomial. $poly(\cdot)$ is recursively defined on $\etree$ as follows, where $\etree_\lchild$ and $\etree_\rchild$ denote the left and right child of $\etree$ respectively.
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@ -92,7 +64,7 @@ $\expandtree{\etree}$ is the pure sum of products expansion of $\etree$. The lo
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\begin{Example}\label{example:expr-tree-T}
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To illustrate with an example, consider the product $(x + 2y)(2x - y)$ and its expression tree $\etree$ in Figure ~\ref{fig:expr-tree-T}. The pure expansion of the product is $2x^2 - xy + 4xy - 2y^2 = \expandtree{\etree}$, logically viewed as $[(2, x^2), (-1, xy), (4, xy), (-2, y^2)]$.
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To illustrate, consider the factorized representation $(x + 2y)(2x - y)$ of the running example. Its expression tree $\etree$ is illustrated in Figure ~\ref{fig:expr-tree-T}. The pure expansion of the product is $2x^2 - xy + 4xy - 2y^2 = \expandtree{\etree}$, logically viewed as $[(2, x^2), (-1, xy), (4, xy), (-2, y^2)]$.
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\end{Example}
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@ -138,7 +110,7 @@ To illustrate with an example, consider the product $(x + 2y)(2x - y)$ and its e
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Let the positive tree, denoted $\abs{\etree}$ be the resulting expression tree such that, for each leaf node $\etree'$ of $\etree$ where $\etree'.\type$ is $\tnum$, $\etree'.\vari{value} = |\etree'.\vari{value}|$. %value $\coef$ of each coefficient leaf node in $\etree$ is set to %$\coef_i$ in $\etree$ is exchanged with its absolute value$|\coef|$.
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\end{Definition}
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Using the same polynomial from the above example, $poly(\abs{\etree}) = (x + 2y)(2x + y) = 2x^2 +xy +4xy + 2y^2 = 2x^2 + 5xy + 2y^2$. Note that this \textit{is not} the same as $\poly(\vct{X})$.
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Using the same factorization from ~\cref{example:expr-tree-T}, $poly(\abs{\etree}) = (x + 2y)(2x + y) = 2x^2 +xy +4xy + 2y^2 = 2x^2 + 5xy + 2y^2$. Note that this \textit{is not} the same as $\poly(\vct{X})$.
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\begin{Definition}[Evaluation]\label{def:exp-poly-eval}
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Given an expression tree $\etree$ and $\vct{v} \in \mathbb{R}^\numvar$, $\etree(\vct{v}) = poly(\etree)(\vct{v})$.
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@ -158,7 +130,7 @@ For any query polynomial $\poly(\vct{X})$, an approximation of $\rpoly(\prob_1,\
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\end{Theorem}
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\subsection{Approximating $\rpoly$}
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We state the approximation algorithm in terms of a $\bi$.
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\subsubsection{Description}
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Algorithm ~\ref{alg:mon-sam} approximates $\rpoly$ using the following steps. First, a call to $\onepass$ on its input $\etree$ produces a non-biased weight distribution over the monomials of $\expandtree{\etree}$ and a correct count of $|\etree|(1,\ldots, 1)$, i.e., the number of monomials in $\expandtree{\etree}$. Next, ~\cref{alg:mon-sam} calls $\sampmon$ to sample one monomial and its sign from $\expandtree{\etree}$. The sampling is repeated $\ceil{\frac{2\log{\frac{2}{\delta}}}{\epsilon^2}}$ times, where each of the samples are evaluated with input $\vct{p}$, multiplied by $1 \times sign$, and summed. The final result is scaled accordingly returning an estimate of $\rpoly$ with the claimed $(\error, \conf)$-bound of ~\cref{lem:mon-samp}.
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@ -3,6 +3,7 @@
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\section{Introduction}
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Modern production databases like Postgres and Oracle use bag semantics. In contrast, most implementations of probabilistic databases (PDBs) are built in the setting of set semantics, where computing the probability of an output tuple is analogous to weighted model counting (a known \sharpphard problem).
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%the annotation of the tuple is a lineage formula ~\cite{DBLP:series/synthesis/2011Suciu}, which can essentially be thought of as a boolean formula. It is known that computing the probability of a lineage formula is \#-P hard in general
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In PDBs, a boolean formula, ~\cite{DBLP:series/synthesis/2011Suciu} also called a lineage formula, encodes the conditions under which each output tuple appears in the result.
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%The marginal probability of this formula being true is the tuple's probability to appear in a possible world.
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@ -99,7 +100,9 @@ Assume the following $\mathbb{B}/\mathbb{N}$ variable assignments: $W_a\mapsto T
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\end{align*}
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In the set/lineage setting, we find that the boolean query is satisfied, while in the bags evaluation we see how many combinations of the input satsify the query.
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\end{Example}
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Note that computing the probability of the query of ~\cref{ex:intro} in set semantics is indeed \sharpphard, since it is a query that is non-hierarchical
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%, i.e., for $Vars(\poly)$ denoting the set of variables occuring across all atoms of $\poly$, a function $sg(x)$ whose output is the set of all atoms that contain variable $x$, we have that $sg(A) \cap sg(B) \neq \emptyset$ and $sg(A)\not\subseteq sg(B)$ and $sg(B)\not\subseteq sg(A)$,
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~\cite{10.1145/1265530.1265571}. %Thus, computing $\expct\pbox{\poly(W_a, W_b, W_c)}$, i.e. the probability of the output with annotation $\poly(W_a, W_b, W_c)$, ($\prob(q)$ in Dalvi, Sucui) is hard in set semantics.
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To see why this computation is hard for query $\poly$ over set semantics, from the query input we compute an output lineage formula of $\poly(W_a, W_b, W_c) = W_aW_b \vee W_bW_c \vee W_cW_a$. Note that the conjunctive clauses are not independent of one another and the computation of the probability is not linear in the size of $\poly(W_a, W_b, W_c)$:
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%root:main.tex
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\subsection{Multiple Distinct $\prob$ Values}
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\section{Multiple Distinct $\prob$ Values}
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We would like to argue for a compressed version of $\poly(\vct{w})$, in general $\expct_{\vct{w}}\pbox{\poly(\vct{w})}$ cannot be computed in linear time.
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\AR{Added the hardness result below.}
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%root: main.tex
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%!TEX root = ./main.tex
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%\onecolumn
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\section{Polynomial Formulation and Equivalences}
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\subsection{Polynomial Formulation and Equivalences}
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Before proceeding, note that the following is assuming $\ti$s in the setting of \textit{bag} semantics.
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Before proceeding, note that the following is assuming $\ti/\bi$ (set) input with the output in the bag setting.
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Throughout the note, we also make the following \textit{assumption}.
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Let us use the expression $(x + y)^2$ for a running example in the following definitions.
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\begin{Definition}[Monomial]\label{def:monomial}
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A monomial is a product of a fixed set of variables, each raised to a non-negative integer power.
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@ -17,14 +17,23 @@ For the term $2xy$, by ~\cref{def:monomial} the monomial is $xy$.
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A polynomial is in standard monomial basis when it is fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
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\end{Definition}
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For example, consider the expression $(x + y)^2$. The standard monomial basis for this expression is $x^2 +2xy + y^2$. While $x^2 + xy + xy + y^2$ is an expanded form of the expression, it is not the standard monomial basis since $xy$ appears more than once.
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The standard monomial basis for the running example is $x^2 +2xy + y^2$. While $x^2 + xy + xy + y^2$ is an expanded form of the expression, it is not the standard monomial basis since $xy$ appears more than once.
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Throughout this paper, we also make the following \textit{assumption}.
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\begin{Assumption}\label{assump:poly-smb}
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All polynomials considered are in standard monomial basis, i.e., $\poly(\vct{X}) = \sum\limits_{\vct{d} \in \mathbb{N}^\numvar}q_d \cdot \prod\limits_{i = 1, d_i \geq 1}^{\numvar}X_i^{d_i}$, where $q_d$ is the coefficient for the monomial encoded in $\vct{d}$ and $d_i$ is the $i^{th}$ element of $\vct{d}$.
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\end{Assumption}
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While the definition of polynomial $\poly(\vct{X})$ over a $\bi$ input doesn't change, we introduce an alternative notation which will come in handy. Given $\ell$ blocks, we write $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
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The number of tuples in the $\bi$ instance can be (trivially) computed as $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .
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\begin{Definition}\label{def:qtilde}
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\begin{Definition}[Degree]\label{def:degree}
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The degree of polynomial $\poly(\vct{X})$ is the maximum sum of the exponents of a monomial, over all monomials when $\poly(\vct{X})$ is in SOP form.
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\end{Definition}
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The degree of the running example is $2$. In this paper we consider only finite degree polynomials.
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\begin{Definition}[$\rpoly(\vct{X})$] \label{def:qtilde}
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Define $\rpoly(X_1,\ldots, X_\numvar)$ as the reduced version of $\poly(X_1,\ldots, X_\numvar)$, of the form
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$\rpoly(X_1,\ldots, X_\numvar) = $
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\end{align*}
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\end{Example}
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Intuitively, $\rpoly(\textbf{X})$ is the expanded sum of products form of $\poly(\textbf{X})$ such that if any $X_j$ term has an exponent $e > 1$, it is reduced to $1$, i.e. $X_j^e\mapsto X_j$ for any $e > 1$.
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Alternatively, one can gain intuition for $\rpoly$ by thinking of $\rpoly$ as the resulting sum of product expansion of $\poly$ when $\poly$ is in a factorized form such that none of its terms have an exponent $e > 1$, if the product operator is idempotent.
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Intuitively, $\rpoly(\textbf{X})$ is the SOP form of $\poly(\textbf{X})$ such that if any $X_j$ term has an exponent $e > 1$, it is reduced to $1$, i.e. $X_j^e\mapsto X_j$ for any $e > 1$.
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Alternatively, one can gain intuition for $\rpoly$ by thinking of $\rpoly$ as the resulting SOP of $\poly(\vct{X})$ with an idemptent product operator.
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The usefulness of this reduction will be seen shortly.
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When considering $\bi$ input, it becomes necessary to redefine $\rpoly(\vct{X})$.
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\begin{Definition}[$\rpoly$ $\bi$ Redefinition]
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A polynomial $\poly(\vct{X})$ over a $\bi$ instance is reduced to $\rpoly(\vct{X})$ with the following criteria. First, all exponents $e > 1$ are reduced to $e = 1$. Second, all monomials sharing the same $\block$ are dropped. Formally this is expressed as
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\begin{equation*}
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\rpoly(\vct{X}) = \poly(\vct{X}) \mod X_i^2 - X_i \mod X_{\block_s, t}X_{\block_s, u}
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\end{equation*}
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for all $i$ in $[\numvar]$ and for all $s$ in $\ell$, such that for all $t, u$ in $[\abs{block_s}]$, $t \neq u$.
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\end{Definition}
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The usefulness of this reduction will be seen in ~\cref{lem:exp-poly-rpoly}.
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\begin{Lemma}\label{lem:pre-poly-rpoly}
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When $\poly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}} \cdot \prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i^{d_i}$, we have then that $\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar} q_{\vct{d}}\cdot\prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i$.
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@ -67,7 +87,7 @@ Note that any $\poly$ in factorized form is equivalent to its sum of product exp
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Define all variables $X_i$ in $\poly$ to be independent.
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\begin{Lemma}\label{lem:exp-poly-rpoly}
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The expectation over possible worlds in $\poly$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
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The expectation over possible worlds in $\poly(\vct{X})$ is equal to $\rpoly(\prob_1,\ldots, \prob_\numvar)$.
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\begin{equation*}
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\expct_{\vct{w}}\pbox{\poly(\vct{w})} = \rpoly(\prob_1,\ldots, \prob_\numvar).
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\end{equation*}
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%root: main.tex
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%!TEX root=./main.tex
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%\onecolumn
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\section{Query translation into polynomials}
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\section{Background Knowledge and Notation}
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\subsection{Introduction}
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\subsection{PDBs}
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An incomplete database $\idb$ is a set of deterministic databases $\db$ where each element is known as a possible world.
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The possible worlds semantics gives a framework for how to think about running queries over $\idb$. Given a query $\query$, $\query$ is deterministically run over each $\db \in \idb$, and the output of $\query(\idb)$ is defined as the set of results (worlds) from running $\query$ over each $\db_i \in \idb$. We write this formally as,
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\[\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}.\]
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\begin{Definition}[$\bi$~\cite{DBLP:series/synthesis/2011Suciu}]
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A Block Independent Database ($\bi$) is a PDB whose tuples are partitioned in blocks, where we denote block $i$ as $\block_i$. Each $\block_i$ is independent of all other blocks, while all tuples sharing the same $\block_i$ are mutually exclusive.
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\end{Definition}
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\begin{Definition}[$\ti$]
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A Tuple Independent Database ($\ti$) is a special case of a $\bi$ such that each tuple is its own block.
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\end{Definition}
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\subsection{Modeling and Semantics}
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Define $\vct{X}$ to be the variables $X_1,\dots,X_M$. We emphasize that formal variables do not have a fixed domain type prior to assignment. Let the set of all tuples in domain of $\sch(\db)$ be $\tset$.
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Define $\vct{X}$ to be the vector of variables $X_1,\dots,X_M$. Let the set of all tuples in domain of $\sch(\db)$ be $\tset$.
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\subsubsection{K-relations}\label{subsubsec:k-rel}
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A K-relation~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are each annotated with an expression whose values come from a commutative K-semiring, denoted $\{K, \oplus, \otimes, \mathbbold{0}, \mathbbold{1}\}$. A commutative $K$-semiring has associative and commutative operators $\oplus$ and $\otimes$, with $\otimes$ distributing over $\oplus$, $\mathbbold{0}$ the identity of $\oplus$, $\mathbbold{1}$ likewise of $\otimes$, and element $\mathbbold{0}$ anihilates all elements of $K$ when being combined with $\otimes$. The information encoded in the annotation depends on the underlying semiring of the relation.
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As noted in \cite{DBLP:conf/pods/GreenKT07}, the $\mathbb{N}[\vct{X}]$-semiring is a semiring over the set $\mathbb{N}[\vct{X}]$ of all polynomials, whose variables can then be substituted with $K$-values from other semirings, evaluating the operators with the operators of the substituted semiring, to produce varying semantics such as set, bag, and security annotations.
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As noted in \cite{DBLP:conf/pods/GreenKT07}, the $\mathbb{N}[\vct{X}]$-semiring is a semiring over the set $\mathbb{N}[\vct{X}]$ of all polynomials, whose variables can then be substituted with $K$-values from other semirings, evaluating the operators with the operators of the substituted semiring, to produce varying semantics such as set, bag, and security.
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Further define $\nxdb$ as an $\mathbb{N}[\vct{X}]$ database where each tuple $\tup \in \db$ is annotated with a polynomial over variables $X_1,\ldots, X_M$.
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Since $\nxdb$ is a database that maps tuples to polynomials, it is customary for arbitrary table $\rel$ to be viewed as a function $\rel: \tset \mapsto \mathbb{N}[\vct{X}]$, where $\rel(\tup)$ denotes the polynomial annotating tuple $\tup$.
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It has been shown in previous work that commutative semirings precisely model translations of RA+ query operations to $K$-annotations.
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The evalution semantics notation $\llbracket \cdot \rrbracket = x$ simply mean that the result of evaluating expression $\cdot$ is given by following the semantics $x$. Given a query $\query$, operations in $\query$ are translated into the following polynomial expressions.
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%The evalution semantics notation $\llbracket \cdot \rrbracket = x$ simply mean that the result of evaluating expression $\cdot$ is given by following the semantics $x$.
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Given a query $\query$, operations in $\query$ are translated into the following polynomial expressions.
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\begin{align*}
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&\eval{\project_A(\rel)}(\tup)&& = &&\sum_{\tup': \project_A(\tup) = \tup} \eval{\rel}(\tup')\\
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@ -40,13 +47,13 @@ The evalution semantics notation $\llbracket \cdot \rrbracket = x$ simply mean t
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&\eval{R}(\tup) && = &&\rel(\tup)
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\end{align*}
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The above semantics show us how to obtain the $K$-annotation on a tuple in the result of query $\query$ from the annotations on the tuples in the input of $\query$. When used with $\mathbb B$-typed variables, an $\mathbb{N}[\vct{X}]$ relation is effectively a C-Table \cite{DBLP:conf/pods/GreenKT07}, since all first order formulas can be equivalently modeled by polynomials, where $\oplus$ is disjunction and $\otimes$ is conjunction.
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Using $\mathbb B$-typed variables in an $\mathbb{N}[\vct{X}]$ relation would correspond to substituting values and operators from the $\{\mathbb{B}, \vee, \wedge, \bot, \top\}$ semiring. In like manner, when using variables from the $\mathbb{N}$ domain, the annotations then effectively model bag semantics, where the variables and $\oplus$ and $\otimes$ operations come from the natural numbers semiring $\{\mathbb{N}, +, \times, 0, 1\}$.
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The above semantics show us how to obtain the $K$-annotation on a tuple in the result of query $\query$ from the annotations of the input tuples. When used with $\mathbb B$-typed variables, an $\mathbb{N}[\vct{X}]$ relation is effectively a C-Table \cite{DBLP:conf/pods/GreenKT07}, since all first order formulas can be equivalently modeled by polynomials, where $\oplus$ is disjunction and $\otimes$ is conjunction.
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This is the equivalent to substituting values and operators from the $\{\mathbb{B}, \vee, \wedge, \bot, \top\}$ semiring. In like manner, when assigning values from the $\mathbb{N}$ domain, the polynomials then model bag semantics, where the variables and $\oplus$ and $\otimes$ operations come from the natural numbers semiring $\{\mathbb{N}, +, \times, 0, 1\}$.
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\subsection{Defining the Data}\label{subsec:def-data}
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For the set of possible worlds, $\wSet$, i.e. the set of all $\db_i \in \idb$, define an injective mapping to the set $\{0, 1\}^M$, where for each vector $\vct{w} \in \{0, 1\}^M$ there is at most one element $\db_i \in \idb$ mapped to $\vct{w}$.
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In the general case, the binary value of $\vct{w}$ uniquely identifies a potential possible world. For example, consider the case of the Tuple Independent Database $(\ti)$ data model in which each table is a set of tuples, each of which is independent of one another, and individually occur with a specific probability $\prob_\tup$. Because of independence, a $\ti$ with $\numvar$ tuples naturally has $2^\numvar$ possible worlds, thus $\numvar = M$, and the injective mapping for each $\vct{w} \in \{0, 1\}^M$ is trivial. In the Block Independent Disjoint data model ($\bi$), because of the disjoint condition on tuples within the same block, a $\bi$ may not have exactly $2^M$ possible worlds since there are combinations of tuples that cannot exist in the encoding.
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Denote a random variable selecting a world according to distribution $P$ to be $\rw$. Provided that for any non-possible world $\vct{w} \in \{0, 1\}^M, \pd[\rw = \vct{w}] = 0$, a probability distribution over $\{0, 1\}^M$ is a distribution over $\Omega$, which we have already defined as $\pd$.
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Denote a random variable selecting a world according to distribution $P$ to be $\rw$. Provided that for any non-possible world $\vct{w} \in \{0, 1\}^M, \pd[\rw = \vct{w}] = 0$, a probability distribution over $\{0, 1\}^M$ is a distribution over $\Omega$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%This could be a way to think of world binary vectors in the general case
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@ -63,6 +70,6 @@ One of the aggregates we desire to compute over the annotated polynomial is the
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Above, $\poly(\vct{w})$ is used to mean the assignment of $\vct{w}$ to $\vct{X}$.
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For a $\ti$, the bit-string world value $\vct{w}$ can be used as indexing to determine which tuples are present in the $\vct{w}$ world, where the $i^{th}$ bit position $(\wbit_i)$ represents whether a tuple $\tup_i$ appears in the unique world identified by the binary value of $\vct{w}[i]$. Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$ such that those probabilities produce the possible worlds in D with a distribution $\pd$ over all worlds. Let $\pd^{(\vct{p})}$ represent the distribution induced by $\vct{p}$.
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For a $\ti$, the bit-string world value $\vct{w}$ can be used as indexing to determine which tuples are present in the $\vct{w}$ world, where the $i^{th}$ bit position $(\wbit_i)$ represents whether a tuple $\tup_i$ appears in the unique world $\vct{w}$. Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$ such that those probabilities produce the possible worlds in D with a distribution $\pd$ over all worlds. Let $\pd^{(\vct{p})}$ represent the distribution induced by $\vct{p}$.
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\[\expct_{\rw\sim \pd^{(\vct{p})}}\pbox{\poly(\rw)} = \sum\limits_{\vct{w} \in \{0, 1\}^\numvar} \poly(\vct{w})\prod_{\substack{i \in [\numvar]\\ s.t. \wElem_i = 1}}\prob_i \prod_{\substack{i \in [\numvar]\\s.t. w_i = 0}}\left(1 - \prob_i\right).\]
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Reference in a new issue