Cleaning Appendix A.
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%!TEX root=./main.tex
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We can use $\semK$-relations to model bags. A \emph{$\semK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = \inset{\domK, \addK, \multK, \zeroK, \oneK}$. A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
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Let $\udom$ be a countable domain of values.
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Formally, an n-ary $\semK$-relation $\rel$ over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$. A $\semK$-database is defined similarly, where we view the $\semK$-database (relation) as a function mapping tuples to their respective annotations.
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@ -18,15 +18,15 @@ To justify the use of $\semNX$-databases, we need to show that we can encode any
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As mentioned above we will use $\semNX$-databases paired with a probability distribution as a representation system, referring to such databases as \abbrNXPDB\xplural.
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Given \abbrNXPDB $\pxdb$, one can think of the of $\pd$ as the probability distribution across all worlds $\inset{0, 1}^\numvar$. Denote a particular world to be $\vct{w}$. For convenience let $\assign_\vct{w}: \pxdb\rightarrow\pndb$ be a function that computes the corresponding $\semN$-\abbrPDB upon assigning all values $w_i \in \vct{w}$ to $X_i \in \vct{X}$ of $\db_{\semNX}$. Note the one-to-one correspondence between elements $\vct{w}\in\inset{0, 1}^\numvar$ to the worlds encoded by $\db_{\semNX}$ when $\vct{w}$ is assigned to $\vct{X}$ (assuming a domain of $\inset{0, 1}$ for each $X_i$).
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We can think of $\assign_\vct{w}(\pxdb)\inparen{\tup}$ as the semiring homomorphism $\semNX \to \semN$ that applies the assignment $\vct{w}$ to all variables $\vct{X}$ of a polynomial and evaluates the resulting expression in $\semN$.
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Given \abbrNXPDB $\pxdb$, one can think of the of $\pd$ as the probability distribution across all worlds $\inset{0, 1}^\numvar$. Denote a particular world to be $\vct{W}$. For convenience let $\assign_\vct{W}: \pxdb\rightarrow\pndb$ be a function that computes the corresponding $\semN$-\abbrPDB upon assigning all values $W_i \in \vct{W}$ to $X_i \in \vct{X}$ of $\db_{\semNX}$. Note the one-to-one correspondence between elements $\vct{W}\in\inset{0, 1}^\numvar$ to the worlds encoded by $\db_{\semNX}$ when $\vct{W}$ is assigned to $\vct{X}$ (assuming a domain of $\inset{0, 1}$ for each $X_i$).
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We can think of $\assign_\vct{W}(\pxdb)\inparen{\tup}$ as the semiring homomorphism $\semNX \to \semN$ that applies the assignment $\vct{W}$ to all variables $\vct{X}$ of a polynomial and evaluates the resulting expression in $\semN$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[$\rmod\inparen{\pxdb}$]\label{def:semnx-pdbs}
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Given an \abbrNXPDB$\pxdb$, we compute its equivalent $\semN$-\abbrPDB $\pndb = \rmod\inparen{\pxdb} = \inparen{\idb, \pd'}$ as:
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\begin{align*}
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\idb & = \{ \assign_{\vct{w}}(\pxdb) \mid \vct{w} \in \{0,1\}^n \} \\
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\forall \db \in \idb: \probOf(\db) & = \sum_{\vct{w} \in \{0,1\}^n: \assign_{\vct{w}}(\pxdb) = \db} \probOf(\vct{w})
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\idb & = \{ \assign_{\vct{W}}(\pxdb) \mid \vct{W} \in \{0,1\}^n \} \\
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\forall \db \in \idb: \probOf(\db) & = \sum_{\vct{W} \in \{0,1\}^n: \assign_{\vct{W}}(\pxdb) = \db} \probOf(\vct{W})
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\end{align*}
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -52,15 +52,16 @@ In $\db_{\semNX}$ we assign each tuple $\tup$ the polynomial:
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\[
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\db_{\semNX}(\tup) = \sum_{i=1}^{\abs{\idb}} D_i(\tup)\cdot X_{i}
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\]
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The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{w}_i$. All elements of $\vct{w}_i$ are zero except for the position corresponding to variables $X_{i}$ which is set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, \abbrNXPDB\xplural are a complete representation system.
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The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{W}_i$. All elements of $\vct{W}_i$ are zero except for the position corresponding to variables $X_{i}$ which is set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, \abbrNXPDB\xplural are a complete representation system.
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Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{w}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{w} := X = 0, Y=1$. We get $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$.
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Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{W}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{W}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{W}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{W}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{W} := X = 0, Y=1$. We get $\assign_\vct{W}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$.
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Closure under $\raPlus$ queries follows from this and from \cite{DBLP:conf/pods/GreenKT07}'s Proposition 3.5, which states that semiring homomorphisms commute with queries over $\semK$-relations.
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\qed
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\end{proof}
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\subsection{\tis and \bis in the \abbrNXPDB model}\label{subsec:supp-mat-ti-bi-def}
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\subsubsection{\tis and \bis in the \abbrNXPDB model}\label{subsec:supp-mat-ti-bi-def}
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Two important subclasses of \abbrNXPDB\xplural that are of interest to us are the bag versions of tuple-independent databases (\tis) and block-independent databases (\bis). Under set semantics, a \ti is a deterministic database $\db$ where each tuple $\tup$ is assigned a probability $\prob_\tup$. The set of possible worlds represented by a \ti $\db$ is all subsets of $\db$. The probability of each world is the product of the probabilities of all tuples that exist with one minus the probability of all tuples of $\db$ that are not part of this world, i.e., tuples are treated as independent random events. In a \bi, we also assign each tuple a probability, but additionally partition $\db$ into blocks. The possible worlds of a \bi $\db$ are all subsets of $\db$ that contain at most one tuple from each block. Note then that the tuples sharing the same block are disjoint, and the sum of the probabilitites of all the tuples in the same block $\block$ is at most $1$.
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The probability of such a world is the product of the probabilities of all tuples present in the world.
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For bag \tis and \bis, we define the probability of a tuple to be the probability that the tuple exists with multiplicity at least $1$.
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@ -68,13 +69,14 @@ For bag \tis and \bis, we define the probability of a tuple to be the probabili
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In this work, we define \tis and \bis as subclasses of \abbrNXPDB\xplural defined over variables $\vct{X}$ (\Cref{def:semnx-pdbs}) where $\vct{X}$ can be partitioned into blocks that satisfy the conditions of a \ti or \bi (stated formally in \Cref{subsec:tidbs-and-bidbs}).
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In this work, we consider one further deviation from the standard: We use bag semantics for queries.
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Even though tuples cannot occur more than once in the input \ti or \bi, they can occur with a multiplicity larger than one in the result of a query.
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Since in \tis and \bis, there is a one-to-one correspondence between tuples in the database and variables, we can interpret a vector $\vct{w} \in \{0,1\}^n$ as denoting which tuples exist in the possible world $\assign_{\vct{w}}(\pxdb)$ (the ones where $w_i = 1$).
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For BIDBs specifically, note that at most one of the bits corresponding to tuples in each block will be set (i.e., for any pair of bits $w_j$, $w_{j'}$ that are part of the same block $b_i \supseteq \{t_{i,j}, t_{i,j'}\}$, at most one of them will be set).
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Since in \tis and \bis, there is a one-to-one correspondence between tuples in the database and variables, we can interpret a vector $\vct{W} \in \{0,1\}^n$ as denoting which tuples exist in the possible world $\assign_{\vct{W}}(\pxdb)$ (the ones where $W_i = 1$).
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For BIDBs specifically, note that at most one of the bits corresponding to tuples in each of the $\numblock$ blocks will be set (i.e., for any pair of bits $W_j$, $W_{j'}$ that are part of the same block $\block_i \supseteq \{t_{j}, t_{j'}\}$, at most one of them will be set).
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Denote the vector $\vct{p}$ to be a vector whose elements are the individual probabilities $\prob_i$ of each tuple $\tup_i$. Given \abbrPDB $\pdb$ where $\pd$ is the distribution induced by $\vct{p}$, which we will denote $\pd^{\inparen{\vct{\prob}}}$.
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%
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\begin{align}\label{eq:tidb-expectation}
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\expct_{\vct{W} \sim \pd^{(\vct{p})}}\pbox{\poly(\vct{W})}
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= \sum\limits_{\substack{\vct{w} \in \{0, 1\}^\numvar\\ s.t. w_j,w_{j'} = 1 \rightarrow \not \exists b_i \supseteq \{t_{i,j}, t_{i,j'}\}}} \poly(\vct{w})\prod_{\substack{j \in [\numvar]\\ s.t. \wElem_j = 1}}\prob_j \prod_{\substack{i\in\pbox{\numblock}~|~\not\exists j\in [\numvar]\\s.t. w_j = 1}}\left(1 - \sum_{j\in\pbox{\numvar}~|~\tup_{i, j} \subseteq\block_i}\prob_i\right)
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= \sum\limits_{\substack{\vct{W} \in \{0, 1\}^\numvar\\ \suchthat W_j,W_{j'} = 1 \rightarrow \not \exists \block_i \supseteq \{t_{j}, t_{j'}\}}} \poly(\vct{W})\prod_{\substack{j \in [\numvar]\\ \suchthat W_j = 1}}\prob_j \prod_{\substack{i\in\pbox{\numblock}\suchthat\\ \forall \tup_j \in \block_i, W_j = 0}}%\not\exists j\in [\numvar]\\\suchthat \tup_{i, j}\subseteq\block_i, W_j = 1}}
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\left(1 - \sum_{\tup_{j} \in \block_i}\prob_j\right)
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\end{align}
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%
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Recall that tuple blocks in a TIDB always have size 1, so the outer summation of \cref{eq:tidb-expectation} is over the full set of vectors.
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@ -87,13 +89,13 @@ We need to prove for $\semN$-PDB $\pdb = (\idb,\pd)$ and \abbrNXPDB $\pxdb = (\d
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By expanding $\nxpolyqdt$ and the expectation we have:
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\begin{align*}
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\expct_{\vct{W} \sim \pd'}\pbox{\poly(\vct{W})}
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& = \sum_{\vct{w} \in \{0,1\}^n}\probOf(\vct{w}) \cdot Q(\db_{\semNX})(t)(\vct{w})\\
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\intertext{From $\rmod(\pxdb) = \pdb$, we have that the range of $\assign_{\vct{w}(\pxdb)}$ is $\idb$, so}
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& = \sum_{\db \in \idb}\;\;\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w}) \cdot Q(\db_{\semNX})(t)(\vct{w})\\
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\intertext{The inner sum is only over $\vct{w}$ where $\assign_{\vct{w}}(\pxdb) = \db$ (i.e., $Q(\db_{\semNX})(t)(\vct{w}) = \query(\db)(t))$}
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& = \sum_{\db \in \idb}\;\;\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w}) \cdot \query(\db)(t)\\
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& = \sum_{\vct{W} \in \{0,1\}^n}\probOf(\vct{W}) \cdot Q(\db_{\semNX})(t)(\vct{W})\\
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\intertext{From $\rmod(\pxdb) = \pdb$, we have that the range of $\assign_{\vct{W}(\pxdb)}$ is $\idb$, so}
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& = \sum_{\db \in \idb}\;\;\sum_{\vct{W} \in \{0,1\}^n : \assign_{\vct{W}}(\pxdb) = \db}\probOf(\vct{W}) \cdot Q(\db_{\semNX})(t)(\vct{W})\\
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\intertext{The inner sum is only over $\vct{W}$ where $\assign_{\vct{W}}(\pxdb) = \db$ (i.e., $Q(\db_{\semNX})(t)(\vct{W}) = \query(\db)(t))$}
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& = \sum_{\db \in \idb}\;\;\sum_{\vct{W} \in \{0,1\}^n : \assign_{\vct{W}}(\pxdb) = \db}\probOf(\vct{W}) \cdot \query(\db)(t)\\
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\intertext{By distributivity of $+$ over $\times$}
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& = \sum_{\db \in \idb}\query(\db)(t)\sum_{\vct{w} \in \{0,1\}^n : \assign_{\vct{w}}(\pxdb) = \db}\probOf(\vct{w})\\
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& = \sum_{\db \in \idb}\query(\db)(t)\sum_{\vct{W} \in \{0,1\}^n : \assign_{\vct{W}}(\pxdb) = \db}\probOf(\vct{W})\\
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\intertext{From the definition of $\pd$ in \cref{def:semnx-pdbs}, given $\rmod(\pxdb) = \pdb$, we get}
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& = \sum_{\db \in \idb}\query(\db)(t) \cdot \probOf(D) \quad = \expct_{\randDB \sim \pd}[\query(\db)(t)]
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\end{align*}
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@ -104,9 +106,9 @@ By expanding $\nxpolyqdt$ and the expectation we have:
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\noindent Note the following fact:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{w}$ such that $\probOf[\vct{W} = \vct{w}] > 0$, it holds that
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\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{W}$ such that $\probOf[\vct{W} = \vct{W}] > 0$, it holds that
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$% \[
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\poly(\vct{w}) = \rpoly(\vct{w}).
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\poly(\vct{W}) = \rpoly(\vct{W}).
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$% \]
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\end{Proposition}
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@ -13,7 +13,8 @@
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\section{Missing details from Section~\ref{sec:background}}\label{sec:proofs-background}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{$\semK$-relations and \abbrNXPDB\xplural}\label{subsec:supp-mat-background}\label{subsec:supp-mat-krelations}
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\subsection{Background details for proof of~\Cref{prop:expection-of-polynom}}
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\subsubsection{$\semK$-relations and \abbrNXPDB\xplural}\label{subsec:supp-mat-background}\label{subsec:supp-mat-krelations}
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\input{app_k-relations}
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\input{app_notation-background}
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@ -198,7 +198,7 @@
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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{\suchthat}{\;s.t.\;} %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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