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@ -140,7 +140,7 @@ level 2/.style={sibling distance=0.7cm},
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\begin{algorithmic}[1]
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\Require \circuit: Circuit
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\Ensure \circuit: Annotated Circuit
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\Ensure \vari{sum} $\in \reals$
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\Ensure \vari{sum} $\in \domR$
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\State $\circuit' \gets \reduce(\circuit)$
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\For{\gate in \topord(\circuit')}\label{alg:one-pass-loop}\Comment{\topord($\cdot$) is the topological order of \circuit}
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\If{\gate.\type $=$ \var}
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@ -5,8 +5,8 @@ The evaluation of $\abs{\circuit}(1,\ldots, 1)$ can be defined recursively, as f
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\begin{align}
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\label{eq:T-all-ones}
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\abs{\circuit}(1,\ldots, 1) = \begin{cases}
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\abs{\circuit_\linput}(1,\ldots, 1) \cdot \abs{\circuit_\rinput}(1,\ldots, 1) &\textbf{if }\circuit.\type = \revision{\circmult}\\
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\abs{\circuit_\linput}(1,\ldots, 1) + \abs{\circuit_\rinput}(1,\ldots, 1) &\textbf{if }\circuit.\type = \revision{\circplus} \\
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\abs{\circuit_\linput}(1,\ldots, 1) \cdot \abs{\circuit_\rinput}(1,\ldots, 1) &\textbf{if }\circuit.\type = \circmult\\
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\abs{\circuit_\linput}(1,\ldots, 1) + \abs{\circuit_\rinput}(1,\ldots, 1) &\textbf{if }\circuit.\type = \circplus \\
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|\circuit.\val| &\textbf{if }\circuit.\type = \tnum\\
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1 &\textbf{if }\circuit.\type = \var.
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\end{cases}
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@ -21,7 +21,7 @@ We now introduce useful definitions and notation related to circuits and polynom
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\begin{Definition}[$\expansion{\circuit}$]\label{def:expand-circuit}
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For a circuit $\circuit$, we define $\expansion{\circuit}$ as a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef \in \reals$.
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For a circuit $\circuit$, we define $\expansion{\circuit}$ as a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef \in \domR$.
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$\expansion{\circuit}$ has the following recursive definition ($\circ$ is list concatenation).
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$\expansion{\circuit} =
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@ -6,10 +6,10 @@
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A \emph{probabilistic database} $\pdb = (\idb, \pd)$ is set of deterministic databases $\idb = \{ \db_1, \ldots, \db_n\}$ called possible worlds, paired with a probability distribution $\pd$ over these worlds.
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A well-studied problem in probabilistic databases is to take a query $\query$ and a probabilistic database $\pdb$, and compute the \emph{marginal probability} of a tuple $\tup$ (i.e., its probability of appearing in the result of query $\query$ over $\pdb$).
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This problem is \sharpphard for set semantics, even for \emph{tuple-independent probabilistic databases}~\cite{DBLP:series/synthesis/2011Suciu} (TIDBs), which are a subclass of probabilistic databases where tuples are independent events. The dichotomy of Dalvi and Suciu~\cite{10.1145/1265530.1265571} separates the hard cases, from cases that are in \ptime for unions of conjunctive queries (UCQs).
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In this work we consider bag semantics, where each tuple is associated with a multiplicity $\db_i(\tup)$ in each possible world $\db_i$ and study the analogous problem of computing the expectation of the multiplicity of a query result tuple $\tup$ (denoted $\query(\db)(t)$):
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In this work we consider bag semantics, where each tuple is associated with a multiplicity $\db_i(\tup)$ in each possible world $\db_i$ and study the analogous problem of computing the expectation (where $\overline{\db}$ denotes a random variable) of the multiplicity of a query result tuple $\tup$ (denoted $\query(\db)(t)$):
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{equation}\label{eq:intro-bag-expectation}
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\expct_{\overline{\mathbf{D}} \sim \probDist}[\query(\overline{\mathbf{D}})(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \pd(\db) \hspace{2cm}\text{\textbf{(Expected Result Multiplicity)}}
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\expct_{\randDB \sim \pd}[\query(\overline{D})(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \probOf\pbox{\db} \hspace{2cm}\text{\textbf{(Expected Result Multiplicity)}}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -130,7 +130,7 @@ A well-known result in probabilistic databases is that under set semantics, the
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over random variables
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($\vct{X}=(X_1,\dots,X_n)$)
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that encode the existence of input tuples. Each possible world $\db$ corresponds to an assignment $\{0,1\}^\numvar$ of the variables in $\vct{X}$ to either true (the tuple exists in this world) or false (the tuple does not exist in this world). Importantly, the following holds: if the lineage formula for $t$ evaluates to true under the assignment for a world $\db$, then $\tup \in \query(\db)$.
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Thus, the marginal probability of tuple $\tup$ is equal to the probability that its lineage evaluates to true (with respect to the obvious analog of probability distribution $\probDist$ defined over $\vct{X}$).
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Thus, the marginal probability of tuple $\tup$ is equal to the probability that its lineage evaluates to true (with respect to the obvious analog of probability distribution $\pd$ defined over $\Omega$ and induced over $\vct{X}$).
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For bag semantics, the lineage of a tuple is a polynomial over variables $\vct{X}=(X_1,\dots,X_n)$ with % \in \mathbb{N}^\numvar$ with
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coefficients in the set of natural numbers $\mathbb{N}$ (an element of semiring $\mathbb{N}[\vct{X}]$).
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@ -140,10 +140,10 @@ In later sections, where we focus on a single lineage polynomial, we will simply
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Example}\label{ex:intro-lineage}
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Associating a lineage variable with every input tuple as shown in \Cref{fig:ex-shipping-simp}, we can compute the lineage of every result tuple as shown in \Cref{subfig:ex-shipping-simp-route}. For example, the tuple Chicago is in the result, because $L_b$ joins with both $R_b$ and $R_c$. Its lineage is $\Phi = L_b \cdot R_b + L_b \cdot R_c$. The expected multiplicity of this result tuple is calculated by summing the multiplicity of the result tuple, weighted by its probability, over all possible worlds.
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Associating a lineage variable with every input tuple as shown in \Cref{fig:ex-shipping-simp}, we can compute the lineage of every result tuple as shown in \Cref{subfig:ex-shipping-simp-queries}. For example, the tuple Chicago is in the result, because $L_b$ joins with both $R_b$ and $R_c$. Its lineage is $\Phi = L_b \cdot R_b + L_b \cdot R_c$. The expected multiplicity of this result tuple is calculated by summing the multiplicity of the result tuple, weighted by its probability, over all possible worlds.
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In this example, $\Phi$ is a sum of products (SOP), and so we can use linearity of expectation to solve the problem in linear time (in the size of $\Phi$).
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The expectation of the sum is the sum of the expectations of monomials.
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The expectation of each monomial is then computed by multiplying the probabilities of the variables (tuples) occurring in the monomial.
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The expectation of each monomial is then computed by multiplying the probabilities of the variables (tuples) occurring in the monomial, since we have independence of tuples.
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The expected multiplicity for Chicago is $1.0$.
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\end{Example}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -187,7 +187,7 @@ The expectation $\expct\pbox{\Phi^2}$ then is:
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\noindent If the domain of a random variable $W$ is $\{0, 1\}$, then for any $k > 0$, $\expct\pbox{W^k} = \expct\pbox{W}$, which means that $\expct\pbox{\Phi^2}$ simplifies to:
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\begin{footnotesize}
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\begin{equation*}
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\expct\pbox{L_a^2}\expct\pbox{L_b} + \expct\pbox{L_b}\expct\pbox{L_d} + \expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_d} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_b}\expct\pbox{L_d}\expct\pbox{L_c}
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\expct\pbox{L_a}\expct\pbox{L_b} + \expct\pbox{L_b}\expct\pbox{L_d} + \expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_d} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_b}\expct\pbox{L_d}\expct\pbox{L_c}
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\end{equation*}
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\end{footnotesize}
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\noindent This property leads us to consider a structure related to the lineage polynomial.
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@ -197,7 +197,7 @@ For any polynomial $\poly(\vct{X})$, define the \emph{reduced polynomial} $\rpol
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With $\Phi^2$ as an example, we have:
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\begin{align*}
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\widetilde{\Phi^2}(L_a, L_b, L_c, L_d)
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=&\; L_aL_b + L_bL_d + L_bW_c + 2L_aL_bL_d + 2L_aL_bL_c + 2L_bL_cL_d
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=&\; L_aL_b + L_bL_d + L_bL_c + 2L_aL_bL_d + 2L_aL_bL_c + 2L_bL_cL_d
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\end{align*}
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It can be verified that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\Phi^2} = \widetilde{\Phi^2}(\probOf\pbox{L_a=1}, \probOf\pbox{L_b=1}, \probOf\pbox{L_c=1}), \probOf\pbox{L_d=1})$). In fact, we show in \Cref{lem:exp-poly-rpoly} that this equivalence holds for {\em all} UCQs over TIDB/BIDB.
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552
macros.tex
552
macros.tex
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@ -1,41 +1,201 @@
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% -*- root: main.tex -*-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% NOTATION
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Circuits
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\newcommand{\caseheading}[1]{\smallskip \noindent \textbf{#1}.~}
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%%%%%
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\newcommand{\wElem}{w} %an element of \vct{w}
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\newcommand{\suchthat}{\;|\;} %such that
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\newcommand{\kElem}{k}%the kth element
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%RA-to-Poly Notation
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\newcommand{\polyinput}[2]{\left(#1,\ldots, #2\right)}
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\newcommand{\numvar}{n}
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\newcommand{\xplural}{s\xspace}
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\xspaceaddexceptions{\xplural}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\BG}[1]{\todo{\textbf{Boris says:$\,$} #1}}
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\newcommand{\SF}[1]{\todo{\textbf{Su says:$\,$} #1}}
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\newcommand{\OK}[1]{\todo[color=gray]{\textbf{Oliver says:$\,$} #1}}
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\newcommand{\AH}[1]{\todo[backgroundcolor=cyan]{\textbf{Aaron says:$\,$} #1}}
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\newcommand{\AR}[1]{\todo[color=green]{\textbf{Atri says:$\,$} #1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% THEOREM LIKE ENVIRONMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%v---what is this?
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\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
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\newtheorem{Theorem}{Theorem}[section]
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\newtheorem{Definition}[Theorem]{Definition}
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\newtheorem{Lemma}[Theorem]{Lemma}
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\newtheorem{Proposition}[Theorem]{Proposition}
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\newtheorem{Corollary}[Theorem]{Corollary}
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\newtheorem{Example}[Theorem]{Example}
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\newtheorem{hypo}[Theorem]{Conjecture}%used in mult_distinct_p.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Rel model
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Need to have all Rel stuff in one place
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\newcommand{\tup}{t}
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\newcommand{\rel}{R}
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\newcommand{\prel}{\mathcal{\rel}}
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\newcommand{\reli}{S}
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\newcommand{\reli}{S}%<----better names?
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\newcommand{\relii}{T}
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\newcommand{\db}{D}
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\newcommand{\idb}{\Omega}
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\newcommand{\pdb}{\mathcal{D}}
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\newcommand{\pxdb}{\mathbf{D}}
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\newcommand{\nxdb}{D(\vct{X})}%\mathbb{N}[\vct{X}] db
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\newcommand{\tset}{\mathcal{T}}%the set of tuples in a database
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\newcommand{\pd}{\vct{P}}%pd for probability distribution
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\newcommand{\eval}[1]{\llbracket #1 \rrbracket}%evaluation double brackets
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\newcommand{\evald}[2]{\eval{{#1}}_{#2}}
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\newcommand{\linsett}[3]{\Phi_{#1,#2}^{#3}}
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\newcommand{\query}{Q}
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\newcommand{\tset}{\mathcal{T}}%the set of tuples in a database
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\newcommand{\join}{\Join}
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\newcommand{\select}{\sigma}
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\newcommand{\project}{\pi}
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\newcommand{\union}{\cup}
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\newcommand{\sch}{sch}
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\newcommand{\attr}[1]{attr\left(#1\right)}
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\newcommand{\rw}{\textbf{W}}%\rw for random world
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% TERMINOLOGY AND ABBREVIATIONS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Perhaps PDB abbreviations should go here?
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\newcommand{\expectProblem}{\textsc{Expected Result Multiplicity Problem}\xspace}
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\newcommand{\termSMB}{standard monomial basis\xspace}
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\newcommand{\abbrSMB}{SMB\xspace}%we already have this; one has to go
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\newcommand{\termSOP}{sum of products\xspace}
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\newcommand{\abbrSOP}{SOP}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Function Names and Typesetting %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\domain}{\func{Dom}}
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\newcommand{\func}[1]{\textsc{#1}}
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\newcommand{\polyf}{\func{poly}}
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\newcommand{\evalmp}{\func{eval}}
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\newcommand{\degree}{\func{deg}}
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\newcommand{\size}{\func{size}}
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\newcommand{\depth}{\func{depth}}
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\newcommand{\topord}{\func{TopOrd}\xspace}
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\newcommand{\smbOf}[1]{\func{\abbrSMB}\inparen{#1}}
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%Verify if we need the above...
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%saving \treesize for now to keep latex from breaking
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\newcommand{\treesize}{\func{size}}
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%I believe this is used in the algo psuedocode
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\newcommand{\sign}{\func{sgn}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% SEMIRINGS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\udom}{\mathcal{U}}
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\newcommand{\domK}{K}
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\newcommand{\semK}{\mathcal{K}}
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\newcommand{\semN}{\mathbb{N}}
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\newcommand{\semNX}{\mathbb{N}[\vct{X}]}
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\newcommand{\onesymbol}{\mathbbold{1}}
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\newcommand{\zerosymbol}{\mathbbold{0}}
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\newcommand{\multsymb}{\otimes}
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\newcommand{\addsymbol}{\oplus}
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\newcommand{\addK}{\addsymbol_{\semK}}
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\newcommand{\multK}{\multsymb_{\semK}}
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\newcommand{\oneK}{\onesymbol_{\semK}}
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\newcommand{\zeroK}{\zerosymbol_{\semK}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Incomplete DB/PDBs %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\idb}{\Omega}
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\newcommand{\pd}{\mathcal{P}}%pd for probability distribution
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\newcommand{\pdb}{\mathcal{D}}
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\newcommand{\pxdb}{\pdb_{\semNX}}%<---changed from the origninal \mathbf{\db}
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\newcommand{\nxdb}{D(\vct{X})}%\mathbb{N}[\vct{X}] db--Are we currently using this?
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%BIDB
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\newcommand{\block}{b}
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\newcommand{\bivar}{x_{\block, i}}
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%PDB Abbreviations
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\newcommand{\abbrPDB}{\textnormal{PDB}\xspace}
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\newcommand{\ti}{TIDB\xspace}
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\newcommand{\tis}{TIDBs\xspace}
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\newcommand{\bi}{BIDB\xspace}
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\newcommand{\bis}{BIDBs\xspace}
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%not sure if we use these; arguably the above abbrev macros should have a name change
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\newcommand{\tiabb}{ti}
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\newcommand{\biabb}{bi}
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\newcommand{\biwset}{\idb_{\biabb}}
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\newcommand{\biord}{\leq_{x_\block}}
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\newcommand{\tiwset}{\idb_{\tiabb}}
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\newcommand{\bipd}{\pd_{\biabb}}
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\newcommand{\tipd}{\pd_{\tiabb}}
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\newcommand{\bipdb}{\pdb_{\biabb}}
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\newcommand{\tipdb}{\pdb_{\tiabb}}
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%--------------------------------
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\newcommand{\probDist}{\vct{\probOf}}%<---I don't think we need this.
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\newcommand{\probAllTup}{\vct{\prob}}%<---I was using simply \vct{\prob}; decide on a convention
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\newcommand{\wSet}{\Omega}%<---We have \idb, the set of possible worlds; decide on one of these
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%Is this being used?
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\newcommand{\pdbx}{X_{DB}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Math Symbols, Functions/Operators, Containers %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Number Sets
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\newcommand{\domR}{\mathbb{R}}%Consider changing this to \domR
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\newcommand{\domN}{\mathbb{N}}
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%Probability, Expectation
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\newcommand{\expct}{\mathop{\mathbb{E}}}%why not just call this \expect
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\newcommand{\probOf}{Pr}%probability function
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%Functions/Operators
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\newcommand{\abs}[1]{\left|#1\right|}
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\newcommand{\suchthat}{\;|\;} %such that
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\newcommand{\comprehension}[2]{\left\{\;#1\;|\;#2\;\right\}}
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\newcommand{\eval}[1]{\llbracket #1 \rrbracket}%evaluation double brackets
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\newcommand{\evald}[2]{\eval{{#1}}_{#2}}
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%Containers
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\newcommand{\pbox}[1]{\left[#1\right]}%<---used for expectation
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\newcommand{\pbrace}[1]{\left\{#1\right\}}
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%consider replacing \pbrace with what is below
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\newcommand{\inparen}[1]{\left({#1}\right)}
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\newcommand{\inset}[1]{\left\{{#1}\right\}}%we already have this as \pbrace; need to pick one
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Variable, Polynomial and Vector Notation
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Instance Variables
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\newcommand{\prob}{p}
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\newcommand{\wElem}{w} %an element of \vct{w}
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%Polynomial Variables
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\newcommand{\pVar}{X}%<----not used but recomment instituting this--pVar for polyVar
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\newcommand{\kElem}{k}%the kth element<---where and how are we using this?
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%Random Variables
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\newcommand{\randWorld}{W}
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\newcommand{\randDB}{\overline{\db}}
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\newcommand{\rvW}{W}%\rvW for random variable of type World
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%One of these needs to go...I think...
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\newcommand{\randomvar}{W}%this little guy needs a home!
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%Container for Polynomial Params
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\newcommand{\polyinput}[2]{\left(#1,\ldots, #2\right)}%do we still use this?
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%Number of Variables--this could easily be number of tups--maybe move to Rel Model?
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\newcommand{\numvar}{n}
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%Vector
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\newcommand{\vct}[1]{{\bf #1}}
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%using \wVec for world bit vector notation<-----Is this still the case?
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%Polynomial
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\newcommand{\poly}{\Phi}
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\newcommand{\polyX}{\poly\inparen{\pVar}}%<---let's see if this proves handy
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\newcommand{\rpoly}{\widetilde{\poly}}%r for reduced as in reduced 'Q'
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\newcommand{\rpolyX}{\rpoly\inparen{\pVar}}%<---if this isn't something we use much, we can get rid of it
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\newcommand{\polyForTuple}{\poly_{\tup}}%do we use this?
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%Do we use this?
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\newcommand{\out}{output}%output aggregation over the output vector
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\newcommand{\prel}{\mathcal{\rel}}%What is this?
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\newcommand{\linsett}[3]{\Phi_{#1,#2}^{#3}}%Where is this used?
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\newcommand{\wbit}{w}%don't think we need this one
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Graph Notation %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\graph}[1]{G^{(#1)}}
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\newcommand{\numocc}[2]{\#\left(#1,#2\right)}
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\newcommand{\eset}[1]{E^{(#1)}_S} %edge set for arbitrary subgraph
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%I don't think we use these anymore
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\newcommand{\linsys}[1]{LS(\graph{#1})}
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\newcommand{\lintime}[1]{LT^{\graph{#1}}}
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\newcommand{\aug}[1]{AUG^{\graph{#1}}}
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@ -43,12 +203,34 @@
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\newcommand{\dtrm}[1]{Det\left(#1\right)}
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\newcommand{\tuple}[1]{\left<#1\right>}
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\newcommand{\indicator}[1]{\onesymbol\inparen{#1}}
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%----------------------------------------------
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Circuit Notation
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\circuit}{\vari{C}}
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\newcommand{\circuitset}[1]{\vari{CSet}\inparen{#1}}
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\newcommand{\circmult}{\times}
|
||||
\newcommand{\circplus}{+}
|
||||
%formally \rchild and \lchild
|
||||
\newcommand{\rinput}{\vari{R}}
|
||||
\newcommand{\linput}{\vari{L}}
|
||||
\newcommand{\inp}{\vari{input}}
|
||||
\newcommand{\inputs}{\vari{inputs}}%do we use this?
|
||||
\newcommand{\subcircuit}{\vari{S}}%does this clash/conflict with \coeffset?
|
||||
\newcommand{\gate}{\vari{g}}
|
||||
\newcommand{\lwght}{\vari{Lweight}}
|
||||
\newcommand{\rwght}{\vari{Rweight}}
|
||||
\newcommand{\prt}{\vari{partial}}
|
||||
\newcommand{\degval}{\vari{degree}}
|
||||
\newcommand{\type}{\vari{type}}
|
||||
%Do we use this?
|
||||
\newcommand{\subgraph}{\vari{S}_{\equivtree(\circuit)}}
|
||||
%-----
|
||||
\newcommand{\cost}{\func{Cost}}
|
||||
\newcommand{\nullval}{NULL}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Query Classes
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\qClass}{\mathcal{Q}}
|
||||
\newcommand{\raPlus}{\ensuremath{\mathcal{RA}^{+}}\xspace}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Datalog
|
||||
|
|
@ -56,9 +238,24 @@
|
|||
\newcommand{\dlImp}[0]{\,\ensuremath{\mathtt{{:}-}}\,}
|
||||
\newcommand{\dlDontcare}{\_}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Query Classes
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\qClass}{\mathcal{Q}}
|
||||
\newcommand{\raPlus}{\ensuremath{\mathcal{RA}^{+}}\xspace}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% COMPLEXITY
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\sharpphard}{\#P-hard\xspace}
|
||||
\newcommand{\sharpwonehard}{\#W[1]-hard\xspace}
|
||||
\newcommand{\ptime}{PTIME\xspace}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Approx Alg
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\randvar}{Y}
|
||||
\newcommand{\coeffset}{S}
|
||||
\newcommand{\distinctvars}{d}
|
||||
|
|
@ -74,6 +271,7 @@
|
|||
\newcommand{\conf}{\delta}
|
||||
|
||||
|
||||
|
||||
%Pseudo Code Notation
|
||||
\newcommand{\plus}{\texttt{+}}
|
||||
\newcommand{\mult}{\texttt{\times}}
|
||||
|
|
@ -81,18 +279,23 @@
|
|||
\newcommand{\approxq}{\algname{Approximate$\rpoly$}}
|
||||
\newcommand{\onepass}{\algname{OnePass}}
|
||||
\newcommand{\sampmon}{\algname{SampleMonomial}}
|
||||
%I don't think we use reduce anymore
|
||||
\newcommand{\reduce}{\algname{Reduce}}
|
||||
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
|
||||
\newcommand{\vari}[1]{\texttt{#1}\xspace}
|
||||
\newcommand{\accum}{\vari{acc}}
|
||||
\newcommand{\numsamp}{\vari{N}}
|
||||
\newcommand{\numedge}{m}
|
||||
\newcommand{\bivec}{\vari{b}_{\vari{vec}}}
|
||||
\newcommand{\numsamp}{\vari{N}}%we have \samplesize above; we can get rid of one of these
|
||||
\newcommand{\numedge}{m}%we have set size above; we can get rid of one of these
|
||||
\newcommand{\bivec}{\vari{b}_{\vari{vec}}}%Section 3--proof in appendix for last theorem
|
||||
|
||||
%Major cleaning needed to get rid of obsolete notation like expression trees, etc.
|
||||
%I don't know that we use any of the expression tree macros anymore; if we do, they would be predominantly in S 3 and 4 and their respective appendices
|
||||
%expression tree T
|
||||
\newcommand{\etree}{\vari{T}}
|
||||
\newcommand{\stree}{\vari{S}}
|
||||
\newcommand{\lchild}{\vari{L}}
|
||||
\newcommand{\rchild}{\vari{R}}
|
||||
%I don't think we talk of T but of C; let's update this. These should be used only in S 2 and S4
|
||||
%members of T
|
||||
\newcommand{\val}{\vari{val}}
|
||||
\newcommand{\wght}{\vari{weight}}
|
||||
|
|
@ -103,77 +306,82 @@
|
|||
%%%%%%%
|
||||
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
|
||||
\renewcommand{\algorithmicensure}{\textbf{Output:}}
|
||||
\newcommand{\smb}{\poly\left(\vct{X}\right)}%smb for standard monomial basis
|
||||
\newcommand{\smbOf}[1]{\textsc{SMB}(#1)}
|
||||
%\newcommand{\smb}{\poly\left(\vct{X}\right)}%smb for standard monomial basis; S 2<---this command is, I believe, unnecessary
|
||||
%not sure if we use this
|
||||
|
||||
%not sure if we use this
|
||||
\newcommand{\etreeset}[1]{\vari{ET}\left(#1\right)}
|
||||
%verify this
|
||||
%\expandtree is a placeholder until I change other files with the new macro name \expansion
|
||||
\newcommand{\expandtree}[1]{\vari{E}(#1)}
|
||||
\newcommand{\expansion}[1]{\vari{E}(#1)}
|
||||
%not sure if we use this; I think the only occurrence would be in the def section of S 4
|
||||
\newcommand{\elist}[1]{\vari{List}\pbox{#1}}
|
||||
%not sure if we use this anymore either
|
||||
\newcommand{\equivtree}{\vari{EET}}
|
||||
|
||||
%expandtree tuple elements:
|
||||
\newcommand{\monom}{\vari{v}}
|
||||
\newcommand{\coef}{\vari{c}}
|
||||
%----------------------------------
|
||||
\newcommand{\abs}[1]{\left|#1\right|}
|
||||
\newcommand{\func}[1]{\textsc{#1}}
|
||||
\newcommand{\polyf}{\func{poly}}
|
||||
\newcommand{\evalmp}{\func{eval}}
|
||||
\newcommand{\degree}{\func{deg}}
|
||||
\newcommand{\size}{\func{size}}
|
||||
\newcommand{\depth}{\func{depth}}
|
||||
\newcommand{\topord}{\func{TopOrd}\xspace}
|
||||
%saving \treesize for now to keep latex from breaking
|
||||
\newcommand{\treesize}{\func{size}}
|
||||
\newcommand{\sign}{\func{sgn}}
|
||||
|
||||
%Random Variable
|
||||
\newcommand{\randomvar}{W}
|
||||
\newcommand{\domain}{\func{Dom}}
|
||||
%PDBs
|
||||
\newcommand{\pdbx}{X_{DB}}
|
||||
\newcommand{\prob}{p}
|
||||
\newcommand{\probOf}{P}
|
||||
\newcommand{\probDist}{\vct{\probOf}}
|
||||
\newcommand{\probAllTup}{\vct{\prob}}
|
||||
\newcommand{\wSet}{\Omega}
|
||||
\newcommand{\ti}{TIDB\xspace}
|
||||
\newcommand{\tis}{TIDBs\xspace}
|
||||
\newcommand{\bi}{BIDB\xspace}
|
||||
\newcommand{\bis}{BIDBs\xspace}
|
||||
\newcommand{\tiabb}{ti}
|
||||
\newcommand{\biabb}{bi}
|
||||
\newcommand{\biwset}{\idb_{\biabb}}
|
||||
\newcommand{\block}{b}
|
||||
\newcommand{\biord}{\leq_{x_\block}}
|
||||
\newcommand{\tiwset}{\idb_{\tiabb}}
|
||||
\newcommand{\bipd}{\pd_{\biabb}}
|
||||
\newcommand{\tipd}{\pd_{\tiabb}}
|
||||
\newcommand{\bipdb}{\pdb_{\biabb}}
|
||||
\newcommand{\bivar}{x_{\block, i}}
|
||||
\newcommand{\tipdb}{\pdb_{\tiabb}}
|
||||
|
||||
% REPRESENTATIONS
|
||||
\newcommand{\rmod}{Mod}
|
||||
\newcommand{\reprs}{\mathcal{M}}
|
||||
|
||||
|
||||
|
||||
% REPRESENTATIONS--this might be Boris' or Atri's stuff; verify if these macros are current
|
||||
\newcommand{\rmod}{Mod}%mod function which transforms N[X]-DB to N-DB (S 2 and App A)
|
||||
\newcommand{\reprs}{\mathcal{M}}%used to define Representation System in App A
|
||||
\newcommand{\repr}{M}
|
||||
|
||||
%Polynomial Reformulation
|
||||
\newcommand{\wbit}{w}
|
||||
\newcommand{\expct}{\mathop{\mathbb{E}}}
|
||||
\newcommand{\pbox}[1]{\left[#1\right]}
|
||||
\newcommand{\pbrace}[1]{\left\{#1\right\}}
|
||||
\newcommand{\vct}[1]{{\bf #1}}
|
||||
%using \wVec for world bit vector notation
|
||||
\newcommand{\poly}{Q}
|
||||
\newcommand{\rpoly}{\widetilde{Q}}%r for reduced as in reduced 'Q'
|
||||
\newcommand{\polyForTuple}{\poly_{\tup}}
|
||||
\newcommand{\out}{output}%output aggregation over the output vector
|
||||
\newcommand{\numocc}[2]{\#\left(#1,#2\right)}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%not sure about these? Perhaps in appendix B for \assign and S 5 for \support?
|
||||
\newcommand{\assign}{\varphi}
|
||||
\newcommand{\support}[1]{supp({#1})}
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\eps}{\epsilon}%<----this is already defined as \error; need to pick one
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Forcing Layouts
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\trimfigurespacing}{\vspace*{-5mm}}
|
||||
\newcommand{\mypar}[1]{\smallskip\noindent\textbf{{#1}.}}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Proof/Section Headings %
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Is this being used?
|
||||
\newcommand{\caseheading}[1]{\smallskip \noindent \textbf{#1}.~}
|
||||
%%%%%
|
||||
|
||||
%%%Adding stuff below so that long chain of display equatoons can be split across pages
|
||||
\allowdisplaybreaks
|
||||
|
||||
%Macro for mult complexity
|
||||
|
||||
\newcommand{\multc}[2]{\overline{\mathcal{M}}\left({#1},{#2}\right)}
|
||||
|
||||
%consider perhaps putting the tikz code into a separate file.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Graph Symbols
|
||||
% Tikz Graph Symbols
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Shift macro
|
||||
\newcommand{\patternshift}[1]{\hspace*{-0.5mm}\raisebox{-0.35mm}{#1}\hspace*{-0.5mm} }
|
||||
|
|
@ -264,172 +472,6 @@
|
|||
\newcommand{\sg}[1]{S^{(#1)}}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% COMMENTS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\newcommand{\BG}[1]{\todo{\textbf{Boris says:$\,$} #1}}
|
||||
\newcommand{\SF}[1]{\todo{\textbf{Su says:$\,$} #1}}
|
||||
\newcommand{\OK}[1]{\todo[color=gray]{\textbf{Oliver says:$\,$} #1}}
|
||||
\newcommand{\AH}[1]{\todo[backgroundcolor=cyan]{\textbf{Aaron says:$\,$} #1}}
|
||||
\newcommand{\SR}[1]{\todo[backgroundcolor=white]{\textbf{Note to self:$\,$} #1}}
|
||||
\newcommand{\AR}[1]{\todo[color=green]{\textbf{Atri says:$\,$} #1}}
|
||||
|
||||
%\newcommand{\AR}[1]{}
|
||||
%\newcommand{\AH}[1]{}
|
||||
%\newcommand{\SF}[1]{}
|
||||
%\newcommand{\BG}[1]{}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Revision Parameters
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\revision}[1]{\color{blue}{#1}\color{black}\xspace}
|
||||
\newcommand{\oldstuff}[1]{\color{gray}{#1}\color{black}\xspace}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Circuit Notation
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\circuit}{\vari{C}}
|
||||
\newcommand{\circuitset}[1]{\vari{CSet}(#1)}
|
||||
\newcommand{\circmult}{\times}
|
||||
\newcommand{\circplus}{+}
|
||||
%formally \rchild and \lchild
|
||||
\newcommand{\rinput}{\vari{R}}
|
||||
\newcommand{\linput}{\vari{L}}
|
||||
\newcommand{\inp}{\vari{input}}
|
||||
\newcommand{\inputs}{\vari{inputs}}
|
||||
\newcommand{\subcircuit}{\vari{S}}
|
||||
\newcommand{\gate}{\vari{g}}
|
||||
\newcommand{\lwght}{\vari{Lweight}}
|
||||
\newcommand{\rwght}{\vari{Rweight}}
|
||||
\newcommand{\prt}{\vari{partial}}
|
||||
\newcommand{\degval}{\vari{degree}}
|
||||
\newcommand{\type}{\vari{type}}
|
||||
\newcommand{\subgraph}{\vari{S}_{\equivtree(\circuit)}}
|
||||
\newcommand{\cost}{\func{Cost}}
|
||||
\newcommand{\nullval}{NULL}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Sets
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\reals}{\mathbb{R}}
|
||||
%
|
||||
%borrowed from Su and Boris; Ihave them here for reference purposes.
|
||||
%needs to be cleaned up
|
||||
%
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% THEOREM LIKE ENVIRONMENTS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
|
||||
|
||||
\newtheorem{Theorem}{Theorem}[section]
|
||||
\newtheorem{Definition}[Theorem]{Definition}
|
||||
\newtheorem{Lemma}[Theorem]{Lemma}
|
||||
\newtheorem{Proposition}[Theorem]{Proposition}
|
||||
\newtheorem{Property}[Theorem]{Property}
|
||||
\newtheorem{Corollary}[Theorem]{Corollary}
|
||||
\newtheorem{Claim}[Theorem]{Claim}
|
||||
\newtheorem{Example}[Theorem]{Example}
|
||||
\newtheorem{Axiom}[Theorem]{Axiom}
|
||||
\newtheorem{Question}[Theorem]{Question}
|
||||
\newtheorem{Assumption}[Theorem]{Assumption}
|
||||
\newtheorem{hypo}[Theorem]{Conjecture}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Rel model
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\newcommand{\tup}{t}
|
||||
\newcommand{\comprehension}[2]{\left\{\;#1\;|\;#2\;\right\}}
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% SEMIRINGS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\udom}{\mathcal{U}}
|
||||
\newcommand{\semN}{\mathbb{N}}
|
||||
\newcommand{\domN}{\mathbb{N}}
|
||||
\newcommand{\semB}{\mathbb{B}}
|
||||
\newcommand{\semNX}{\mathbb{N}[\vct{X}]}
|
||||
\newcommand{\domK}{K}
|
||||
\newcommand{\semK}{\mathcal{K}}
|
||||
|
||||
\newcommand{\support}[1]{supp({#1})}
|
||||
|
||||
\newcommand{\onesymbol}{\mathbbold{1}}
|
||||
\newcommand{\zerosymbol}{\mathbbold{0}}
|
||||
\newcommand{\multsymb}{\otimes}
|
||||
\newcommand{\addsymbol}{\oplus}
|
||||
\newcommand{\addormultsymbol}{\bigcdot}
|
||||
\newcommand{\monsymbol}{\ominus}
|
||||
|
||||
\newcommand{\addK}{\addsymbol_{\semK}}
|
||||
\newcommand{\multK}{\multsymb_{\semK}}
|
||||
\newcommand{\genopK}{\odot_{\semK}}
|
||||
\newcommand{\monK}{\monsymbol_{\semK}}
|
||||
\newcommand{\oneK}{\onesymbol_{\semK}}
|
||||
\newcommand{\zeroK}{\zerosymbol_{\semK}}
|
||||
\newcommand{\addOf}[1]{\addsymbol_{#1}}
|
||||
\newcommand{\multOf}[1]{\multsymb_{#1}}
|
||||
\newcommand{\monOf}[1]{\monsymbol_{#1}}
|
||||
\newcommand{\oneOf}[1]{\onesymbol_{#1}}
|
||||
\newcommand{\zeroOf}[1]{\zerosymbol_{#1}}
|
||||
|
||||
\newcommand{\dbDomK}[1]{\mathcal{DB}_{#1}}
|
||||
|
||||
\newcommand{\assign}{\varphi}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% COMPLEXITY
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\sharpphard}{\#P-hard\xspace}
|
||||
\newcommand{\sharpwonehard}{\#W[1]-hard\xspace}
|
||||
\newcommand{\ptime}{PTIME\xspace}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TERMINOLOGY AND ABBREVIATIONS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\expectProblem}{\textsc{Expected Result Multiplicity Problem}\xspace}
|
||||
\newcommand{\termSMB}{standard monomial basis\xspace}
|
||||
\newcommand{\abbrSMB}{SMB\xspace}
|
||||
\newcommand{\termSOP}{sum of products\xspace}
|
||||
\newcommand{\abbrSOP}{SOP}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\eps}{\epsilon}
|
||||
\newcommand{\inparen}[1]{\left({#1}\right)}
|
||||
\newcommand{\inset}[1]{\left\{{#1}\right\}}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Forcing Layouts
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\trimfigurespacing}{\vspace*{-5mm}}
|
||||
\newcommand{\mypar}[1]{\smallskip\noindent\textbf{{#1}.}}
|
||||
|
||||
%%%Adding stuff below so that long chain of display equatoons can be split across pages
|
||||
\allowdisplaybreaks
|
||||
|
||||
%Macro for mult complexity
|
||||
|
||||
\newcommand{\multc}[2]{\overline{\mathcal{M}}\left({#1},{#2}\right)}
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "main"
|
||||
|
|
|
|||
|
|
@ -6,19 +6,19 @@
|
|||
We now introduce some terminology % for polynomials
|
||||
and develop a reduced form (a closed form of the polynomial's expectation) for polynomials over probability distributions derived from a \bi or \ti.
|
||||
%We will use $(X + Y)^2$ as a running example.
|
||||
Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ is formally defined as:
|
||||
Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ is formally defined as (with $c_\vct{i} \in \domN$):
|
||||
\begin{equation}
|
||||
\label{eq:sop-form}
|
||||
Q(X_1,\dots,X_n)=\sum_{\vct{d}=(d_1,\dots,d_n)\in \semN^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i}.
|
||||
\poly\inparen{X_1,\dots,X_n}=\sum_{\vct{d}=(d_1,\dots,d_n)\in \semN^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i}.
|
||||
\end{equation}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Standard Monomial Basis]\label{def:smb}
|
||||
From above, the term $\prod_{i=1}^n X_i^{d_i}$ is a {\em monomial}. A polynomial $Q(\vct{X})$ is in standard monomial basis (SMB) when we keep only the terms with $c_{\vct{i}}\ne 0$ from \Cref{eq:sop-form}.
|
||||
From above, the term $\prod_{i=1}^n X_i^{d_i}$ is a {\em monomial}. A polynomial $\poly\inparen{\vct{X}}$ is in standard monomial basis (\abbrSMB) when we keep only the terms with $c_{\vct{i}}\ne 0$ from \Cref{eq:sop-form}.
|
||||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
We consider \abbrSMB as the default representation of a polynomial.
|
||||
We use $\smbOf{\poly}$ to denote the SMB form of a polynomial $\poly$.
|
||||
We use $\smbOf{\poly}$ to denote the \abbrSMB form of a polynomial $\poly$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Degree]\label{def:degree}
|
||||
|
|
@ -29,21 +29,7 @@ The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such
|
|||
The degree of the polynomial $X^2+2XY+Y^2$ is $2$.
|
||||
Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins in any clause of the UCQ query that created it.
|
||||
In this paper we consider only finite degree polynomials.
|
||||
We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if
|
||||
%\AH{Why is it required for the tuple to be n-ary? I think this slightly confuses me since we have n tuples.}
|
||||
% OK: agreed w/ AH, this can be treated as implicit
|
||||
there exists a $\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\query(\vct{X}) = \query(\pxdb)(\tup)$. % Before proceeding, note that the following is assume that polynomials are \bis (which subsume \tis as a special case).
|
||||
|
||||
%\SF{Where is $\block_{i, j}$ used? Is it $X_{\block_{1, 1}}$ or $X_{\block_1, 1}$ ?}
|
||||
% and the probability distribution of $\pxdb$ is uniquely determined based on a probability vector $\vct{p}$ that associates each tuple a probability
|
||||
% variables are independent of each other (or disjoint if they are from the same block) and each variable $X$ is associated with a probability $\vct{p}(X) = \pd[X = 1]$. Thus, we are dealing with polynomials $\poly(\vct{X})$ that are annotations of a tuple in the result of a query $\query$ over a BIDB $\pxdb$ where $\vct{X}$ is the set of variables that occur in annotations of tuples of $\pxdb$.
|
||||
|
||||
% 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 $[\abs{\block_i}]$.
|
||||
% The number of tuples in the $\bi$ instance can be (trivially) computed as $\numvar = \sum\limits_{i = 1}^{\ell}\abs{\block_i}$ .
|
||||
|
||||
|
||||
|
||||
|
||||
We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a \AH{Which formalism? UCQ?}$\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\poly\inparen{\vct{X}} = \query(\pxdb)(\tup)$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Modding with a set]\label{def:mod-set}
|
||||
|
|
@ -126,7 +112,7 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Valid Worlds]
|
||||
For probability distribution $\probDist$, % and its corresponding probability mass function $\probOf$,
|
||||
the set of valid worlds $\eta$ consists of all the worlds with probability value greater than $0$; i.e., for variable vector $\vct{W}$
|
||||
the set of valid worlds $\eta$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
|
||||
\[
|
||||
\eta = \comprehension{\vct{w}}{\probOf[\vct{W} = \vct{w}] > 0}
|
||||
\]
|
||||
|
|
|
|||
|
|
@ -11,7 +11,7 @@ We represent query polynomials via {\em arithmetic circuits}~\cite{arith-complex
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Circuit]\label{def:circuit}
|
||||
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source nodes (in degree of $0$) consist of elements in either $\reals$ or $\vct{X}$. The internal nodes and (the single) sink node of $\circuit$ (corresponding to the result tuple $t$) have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
|
||||
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source nodes (in degree of $0$) consist of elements in either $\domR$ or $\vct{X}$. The internal nodes and (the single) sink node of $\circuit$ (corresponding to the result tuple $t$) have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
|
||||
%
|
||||
Each node in a circuit $\circuit$ has the following members: \type, \val, \vpartial, \vari{input}, \degval and \vari{Lweight}, \vari{Rweight}, where \type is the type of value stored in the node (one of $\{\circplus, \circmult, \var, \tnum\}$, \val is the value stored (a constant or variable), and \vari{input} is the list of the nodes inputs. We use $\circuit_\linput$ to denote the left input and $\circuit_\rinput$ the right input or the sink of circuit $\circuit$.
|
||||
%The member \degval holds the degree of \circuit.
|
||||
|
|
@ -108,13 +108,12 @@ Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corr
|
|||
Note that $\circuit$ need not encode an expression in SMB. For instance, $\circuit$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$, as shown in \Cref{fig:circuit}, while $\polyf(\circuit) = 2X^2+3XY-2Y^2$.
|
||||
|
||||
\begin{Definition}[Circuit Set]\label{def:circuit-set}
|
||||
$\circuitset{\smb}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \poly(\vct{X})$.
|
||||
$\circuitset{\smbOf{\polyX}}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \smbOf\polyX$.
|
||||
\end{Definition}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. One can think of $\circuitset{\smb}$ as the infinite set of circuits each of which model an encoding (factorization) equal to $\polyf(\circuit)$.
|
||||
%\supset \{2X^2 + 3XY - 2Y^2, (X + 2Y)(2X - Y), X(2X - Y) + 2Y(2X - Y), 2X(X + 2Y) - Y(X + 2Y)\}$.
|
||||
The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. One can think of $\circuitset{\smbOf{\polyX}}$ as the infinite set of circuits each of which model an encoding (factorization) equal to $\polyf(\circuit)$.
|
||||
Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\circuit)}$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
@ -126,7 +125,7 @@ Note that \Cref{def:circuit-set} implies that $\circuit \in \circuitset{\polyf(\
|
|||
Let $\vct{X} = (X_1, \ldots, X_n)$, and $\pdb$ be an $\semNX$-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X} \to \{0,1\}$, $\query$ an n-ary query, and $t$ an n-ary tuple.
|
||||
The \expectProblem is defined as follows:\\[-7mm]
|
||||
\begin{center}
|
||||
\textbf{Input}: A circuit $\circuit \in \circuitset{\smb}$ for $\poly(\vct{X}) = \query(\pxdb)(t)$
|
||||
\textbf{Input}: A circuit $\circuit \in \circuitset{\smbOf{\polyX}}$ for $\poly(\vct{X}) = \query(\pxdb)(t)$
|
||||
\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
|
||||
\end{center}
|
||||
\end{Definition}
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@
|
|||
\section{Background and Notation}\label{sec:background}
|
||||
|
||||
\iffalse
|
||||
\subsection{Superlinearity of Bag PDBs}\label{sec:suplin-bags}
|
||||
\subsection{Superlinearity of Bag \abbrPDB\xplural}\label{sec:suplin-bags}
|
||||
Moving forward, we focus exclusively on bags. For $Q()\dlImp$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$ $OnTime(\text{City}')$ over the bag relations of \Cref{fig:ex-shipping-simp}, consider the product query $\poly^2()\dlImp Q \times Q$.
|
||||
The factorized representation of $\poly^2$ is (for simplicity we ignore the random variables of $Route$ since each variable has probability of $1$):
|
||||
\begin{equation*}
|
||||
|
|
@ -33,18 +33,18 @@ With $\poly^2$ as an example, we have:
|
|||
It can be verified that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(\probOf\pbox{L_a=1}, \probOf\pbox{L_b=1}, \probOf\pbox{L_c=1}), \probOf\pbox{L_d=1})$).
|
||||
|
||||
The reduced form of a lineage polynomial can be obtained but requires a linear scan over the clauses of an SOP encoding of the polynomial. Note that for a compressed representation, this scheme would require an exponential number of computations in the size of the compressed representation. In \Cref{sec:hard}, we use $\rpoly$ to prove our hardness results .
|
||||
%In prior work on lineage-based Bag-PDBs~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,yang:2015:pvldb:lenses} where this encoding is implicitly assumed, computing the expected count is linear in the size of the encoding.
|
||||
%In prior work on lineage-based Bag-\abbrPDB\xplural~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,yang:2015:pvldb:lenses} where this encoding is implicitly assumed, computing the expected count is linear in the size of the encoding.
|
||||
%In general however, compressed encodings of the polynomial can be exponentially smaller in $k$ for $k$-products --- the query $\poly^k$ obtained by taking the product of $k$ copies of $\poly$ as a factorized encoding of size $6\cdot k$, while the SOP encoding is of size $2\cdot 3^k$.
|
||||
%This leads us to the \textbf{central question of this paper}:
|
||||
%\begin{quote}
|
||||
%{\em
|
||||
%Is it always the case that the expectation of a UCQ in a Bag-PDB can be computed in time linear in the size of the \textbf{compressed} lineage polynomial?}
|
||||
%Is it always the case that the expectation of a UCQ in a Bag-\abbrPDB can be computed in time linear in the size of the \textbf{compressed} lineage polynomial?}
|
||||
%\end{quote}
|
||||
%If so, then Bag-PDBs can indeed compete with deterministic databases.
|
||||
%If so, then Bag-\abbrPDB\xplural can indeed compete with deterministic databases.
|
||||
%This is unfortunately not the case, and an approximation is required.
|
||||
\fi
|
||||
|
||||
\subsection{Probabilistic Databases (PDBs)}
|
||||
\subsection{Probabilistic Databases (\abbrPDB\xplural)}
|
||||
|
||||
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 the 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: $\query(\idb) = \comprehension{\query(\db)}{\db \in \idb}$.
|
||||
|
|
@ -53,12 +53,12 @@ For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is th
|
|||
%
|
||||
\[\forall \db \in \query(\idb): \pd'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \pd(\db') \]
|
||||
|
||||
Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_n)$ with natural number coefficients and exponents.
|
||||
Let $\semNX$ denote the set of polynomials over variables $\vct{X}=(X_1,\dots,X_\numvar)$ with natural number coefficients and exponents.
|
||||
We model incomplete relations using Green et. al.'s $\semNX$-databases~\cite{DBLP:conf/pods/GreenKT07}, discussed in detail in \Cref{subsec:supp-mat-krelations}. % and summarized here.
|
||||
$\semNX$-relations are functions from tuples to elements of $\semNX$, typically called annotations.
|
||||
We write $R(t)$ to denote the polynomial annotating tuple $t$ in relation $R$. Note that $R(t)$ is the lineage polynomial for $t$.
|
||||
Each possible world is defined by an assignment of $N$ binary values $\vct{W} \in \{0, 1\}^{\abs{\vct{X}}}$ to $\vct{X}$.
|
||||
The multiplicity of $t \in R$ in this possible world, denoted $R(t)(\vct{W})$, is obtained by evaluating the polynomial annotating $t$ on $\vct{W}$.
|
||||
Each possible world is defined by an assignment of $\numvar$ binary values $\vct{\wElem} \in \{0, 1\}^{\numvar}$ to $\vct{X}$.
|
||||
The multiplicity of $t \in R$ in this possible world, denoted $R(t)(\vct{\wElem})$, is obtained by evaluating the polynomial annotating $t$ on $\vct{\wElem}$.
|
||||
$\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
|
||||
|
||||
|
||||
|
|
@ -88,12 +88,12 @@ $\semNX$-relations are closed under $\raPlus$ (\Cref{fig:nxDBSemantics}).
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
We will use $\semNX$-PDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$.
|
||||
We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-PDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{|\vct X|} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
|
||||
$\semNX$-PDBs and a function $\rmod$ (which transforms an $\semNX$-PDB to classical, or $\semN$-PDB~\cite{DBLP:conf/pods/GreenKT07,feng:2019:sigmod:uncertainty}) are both formalized in \Cref{subsec:supp-mat-background}.
|
||||
We will use $\semNX$-\abbrPDB $\pxdb$, defined as the tuple $(\idb_{\semNX}, \pd)$, where $\semNX$-database $\idb_{\semNX}$ is paired with probability distribution $\pd$.
|
||||
We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ on an implicit $\semNX$-\abbrPDB (i.e., $\polyForTuple = \query(\pxdb)(t)$ for some $\pxdb$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{\numvar} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
|
||||
$\semNX$-\abbrPDB\xplural and a function $\rmod$ (which transforms an $\semNX$-\abbrPDB to a classical bag-\abbrPDB, or $\semN$-\abbrPDB~\cite{DBLP:conf/pods/GreenKT07,feng:2019:sigmod:uncertainty}) are both formalized in \Cref{subsec:supp-mat-background}.
|
||||
\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})}. $
|
||||
Given an $\semN$-\abbrPDB $\pdb = (\idb,\pd)$ and $\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$, we have:
|
||||
$ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\randWorld\sim \pd'}\pbox{\polyForTuple(\randWorld)}. $
|
||||
\footnote{Although assumed by most prior work on set-probabilistic databases, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
|
||||
\end{Proposition}
|
||||
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.
|
||||
|
|
@ -102,15 +102,15 @@ We focus on this problem from now on, assume an implicit result tuple, and so dr
|
|||
|
||||
\subsubsection{\tis and \bis}
|
||||
\label{subsec:tidbs-and-bidbs}
|
||||
In this paper, we focus on two popular forms of PDBs: Block-Independent (\bi) and Tuple-Independent (\ti) PDBs.
|
||||
In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
|
||||
%
|
||||
A \bi $\pxdb = (\idb_{\semNX}, \pd)$ is an $\semNX$-PDB such that (i) every tuple is annotated with either $0$ (i.e., the tuple does not exist) or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pxdb$ for which $\pxdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same block are disjoint events.
|
||||
A \bi $\pxdb = (\idb_{\semNX}, \pd)$ is an $\semNX$-\abbrPDB such that (i) every tuple is annotated with either $0$ (i.e., the tuple does not exist) or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pxdb$ for which $\pxdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same block are disjoint events.
|
||||
In other words, each random variable corresponds to the event of a single tuple's presence.
|
||||
%
|
||||
A \emph{\ti} is a \bi where each block contains exactly one tuple.
|
||||
\Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
|
||||
%
|
||||
In a \bi (and by extension a \ti) $\pxdb$, tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \pd[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
|
||||
In a \bi (and by extension a \ti) $\pxdb$, tuples are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_{i,j} \in \block_i$ is associated with a probability $\prob_{\tup_{i,j}} = \probOf[X_{i,j} = 1]$, and is annotated with a unique variable $X_{i,j}$.\footnote{
|
||||
Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function). For $t_j \in b_i$, the event $(X_{i,j} = 1)$ corresponds to the event $(X_i = j)$ in the customary annotation scheme.
|
||||
}
|
||||
Because blocks are independent and tuples from the same block are disjoint, the probabilities $\prob_{\tup_{i,j}}$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
|
||||
|
|
|
|||
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Reference in New Issue