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\section{Introduction}
\label{sec:intro}

As explainability and fairness become more relevant to the data science community, it is now more critical than ever to understand how reliable a dataset is.
Probabilistic databases (PDBs)~\cite{DBLP:series/synthesis/2011Suciu} are a compelling solution, but a major roadblock to their adoption remains:
PDBs are orders of magnitude slower than classical (i.e., deterministic) database systems~\cite{feng:2019:sigmod:uncertainty}.
Naively, one might suggest that this is because most work on probabilistic databases assumes set semantics, while, virtually all implementations of the relational data model use bag semantics.
However, as we show in this paper, there is a more subtle problem behind this barrier to adoption.

A fundamental problem in probabilistic query processing is: given a query, probabilistic database, and possible result tuple, compute the marginal probability of the tuple appearing in the result.
In the set semantics setting, it was shown that this is equivalent to computing the probability of a Boolean formula called the lineage formula, which records how the result tuple was derived from input tuples. 
Given this correspondence, the problem reduces to weighted model counting over the lineage (a \sharpphard problem, even if the lineage is in DNF). 
A large body of work has focused on identifying tractable cases by either identifying tractable classes of queries (e.g.,~\cite{DS12}) or studying compressed representations of lineage formulas that are tractable for certain classes of input databases (e.g.,~\cite{AB15}).  In this work we define a compressed representation as any one of the possible circuit representations of the lineage formula (please see Definitions~\ref{def:circuit},~\ref{def:poly-func}, and~\ref{def:circuit-set}).

The problem of computing the marginal probability of a result tuple has a natural correspondence under bag semantics: computing the expected multiplicity of a query result tuple.
Analogously, this problem can be reduced to computing the expectation of the lineage, which under bag semantics is the standard polynomial parameterized in the lineage variables, with the usual notion of mulitplication and addition operations. 
This problem has received much less attention, perhaps because the problem is trivially tractable.
In fact it is linear time when the lineage polynomial is encoded in the typical sum of products (SOP) representation. 
However, there exist compressed representations of polynomials, e.g., factorizations~\cite{factorized-db}, that can be polynomially more concise than the SOP representation of a polynomial. 
These compression schemes are analogous to typical database optimizations like projection push-down~\cite{DBLP:books/daglib/0020812}, (where e.g. in the case of a projection followed by a join, addition would be performed prior to multiplication, yielding a product of sums), hinting that perhaps even Bag-PDBs have higher query processing complexity than deterministic databases.
In this paper, we confirm this intuition, first proving (by reduction from counting $k$-matchings) that computing the expected count of a query result tuple is super-linear (\sharpwonehard) in the size of a compressed lineage representation, and then relating the size of the compressed lineage to the cost of answering a deterministic query.

In spite of this negative result, not everything is lost. 
We develop an approximation algorithm for expected counts of SPJU query results over Bag-PDBs that is, to our knowledge, the first linear time (in the size of the factorized lineage) $(1-\epsilon)$-approximation.
By extension, this algorithm only has a constant factor slower runtime relative to deterministic query processing.\footnote{
	Monte-carlo sampling~\cite{jampani2008mcdb} is also trivially a constant factor slower, but can only guarantee additive rather than our stronger multiplicative bounds.
} 
This is an important result, because it implies that computing approximate expectations for SPJU queries can indeed be competitive with deterministic query evaluation over bag databases.


\subsection{Sets vs Bags}

\begin{Example}\label{ex:intro}
The tables $\rel$ and $E$ in \Cref{fig:intro-ex} are examples of an incomplete database.  Every tuple $\tup$ (disregard $\Phi_{bag}$ for the moment) of these tables is annotated with a variable or the symbol $\top$.  Each assignment of values to variables ($\{\;W_a,W_b,W_c\;\}\mapsto \{\;\top,\bot\;\}$) identifies one \emph{possible world}, a deterministic database instance containing exactly the tuples annotated by the constant $\top$ or by a variable assigned to $\top$.  When each variable represents an \emph{independent} event, this encoding is called a Tuple Independent Database $(\ti)$.

The probability of this world is the joint probability of the corresponding assignments.
For example, let $\probOf[W_a] = \probOf[W_b] = \probOf[W_c] = \prob$ and consider the possible world where $R = \{\;\tuple{a}, \tuple{b}\;\}$.
The corresponding variable assignment is $\{\;W_a \mapsto \top, W_b \mapsto \top, W_c \mapsto \bot\;\}$, and its probability is $\probOf[W_a]\cdot \probOf[W_b] \cdot \probOf[\neg W_c] = \prob\cdot \prob\cdot (1-\prob)=\prob^2-\prob^3$.
\end{Example}

\begin{figure}[t]
	\begin{subfigure}{0.33\linewidth}
		\centering
		\resizebox{!}{10mm}{
		\begin{tabular}{ c | c c c}
			$\rel$ & A & $\Phi_{set}$ & $\Phi_{bag}$\\
			\hline
			& a & $W_a$ & $W_a$\\
			& b & $W_b$ & $W_b$\\
			& c & $W_c$ & $W_c$\\
		\end{tabular}
}		\caption{Relation $R$ in ~\Cref{ex:intro}}
		\label{subfig:ex-atom1}
	\end{subfigure}%
	\begin{subfigure}{0.33\linewidth}
		\centering
		\resizebox{!}{10mm}{
		\begin{tabular}{ c | c c c c}
			$E$ & A & B & $\Phi_{set}$ & $\Phi_{bag}$ \\
			\hline
			& a & b & $\top$ & $1$\\
			& b & c & $\top$ & $1$\\
			& c & a & $\top$ & $1$\\
		\end{tabular}
		}
		\caption{Relation $E$ in ~\Cref{ex:intro}}
		\label{subfig:ex-atom3}
	\end{subfigure}%
	\begin{subfigure}{0.33\linewidth}
		\centering
	\resizebox{!}{29mm}{
		\begin{tikzpicture}[thick]
			\node[tree_node] (a1) at (0, 0){$W_a$};
			\node[tree_node] (b1) at (1, 0){$W_b$};
			\node[tree_node] (c1) at (2, 0){$W_c$};
			\node[tree_node] (d1) at (3, 0){$W_d$};
	
			\node[tree_node] (a2) at (0.75, 0.8){$\boldsymbol{\circmult}$};
			\node[tree_node] (b2) at (1.5, 0.8){$\boldsymbol{\circmult}$};
			\node[tree_node] (c2) at (2.25, 0.8){$\boldsymbol{\circmult}$};
	
			\node[tree_node] (a3) at (1.9, 1.6){$\boldsymbol{\circplus}$};
			\node[tree_node] (a4) at (0.75, 1.6){$\boldsymbol{\circplus}$};
			\node[tree_node] (a5) at (0.75, 2.5){$\boldsymbol{\circmult}$};
	
			\draw[->] (a1) -- (a2);
			\draw[->] (b1) -- (a2);
			\draw[->] (b1) -- (b2);
			\draw[->] (c1) -- (b2);
			\draw[->] (c1) -- (c2);
			\draw[->] (d1) -- (c2);
			\draw[->] (c2) -- (a3);
			\draw[->] (a2) -- (a4);
			\draw[->] (b2) -- (a3);
			\draw[->] (a3) -- (a4);
			%sink
			\draw[thick, ->] (a4.110) -- (a5.250);
			\draw[thick, ->] (a4.70) -- (a5.290);
			\draw[thick, ->] (a5) -- (0.75, 3.0);
		\end{tikzpicture}
	}
	\caption{Circuit encoding for query $\poly^2$.}
	\label{fig:circuit-q2-intro}
	\end{subfigure}
	%\vspace*{3mm}
  \vspace*{-3mm}
	\caption{ }%{$\ti$ relations for $\poly$}
	\label{fig:intro-ex}
  \trimfigurespacing
\end{figure}


Following prior efforts~\cite{feng:2019:sigmod:uncertainty,DBLP:conf/pods/GreenKT07,GL16}, we generalize this model of Set-PDBs to bags using $\semN$-valued random variables (i.e., $\domain(\randomvar_i) \subseteq \mathbb N$) and constants (annotation $\Phi_{bag}$ in the example).
Without loss of generality, we assume that input relations are sets (i.e. $Dom(W_i) = \{0, 1\}$), while query evaluation follows bag semantics.

\begin{Example}\label{ex:bag-vs-set}
Continuing the prior example, we are given the following Boolean (resp,. count) query
$$\poly() :- R(A), E(A, B), R(B)$$
The lineage of the result in a Set-PDB (resp., Bag-PDB) is a Boolean formula  (resp., polynomial) over random variables annotating the input relations (i.e., $W_a$, $W_b$, $W_c$).
Because the query result is a nullary relation, we write $Q(\cdot)$ to denote the function that evaluates the lineage over one specific assignment of values to the variables (i.e., the value of the lineage in the corresponding possible world):

\setlength\parindent{0pt}
\vspace*{-3mm}
\begin{tabular}{@{}l l}
	\begin{minipage}[b]{0.45\linewidth}
		\begin{equation}
			\poly_{set}(W_a, W_b, W_c) = W_aW_b \vee W_bW_c \vee W_cW_a\label{eq:poly-set}
		\end{equation}
	\end{minipage}\hspace*{5mm}
	&
	\begin{minipage}[b]{0.45\linewidth}
		\begin{equation}
			\poly_{bag}(W_a, W_b, W_c) = W_aW_b + W_bW_c + W_cW_a\label{eq:poly-bag}
		\end{equation}
	\end{minipage}\\
\end{tabular}
\vspace*{1mm}



These functions compute the existence (resp., count) of the nullary tuple resulting  from applying $\poly$ on the PDB of \Cref{fig:intro-ex}.
For the same possible world as in the prior example:
$$
\begin{tabular}{c c}
	\begin{minipage}[b]{0.45\linewidth}
	$\poly_{set}(\top, \top, \bot) = \top\top \vee \top\bot \vee \top\bot = \top$
	\end{minipage}
	&
	\begin{minipage}[b]{0.45\linewidth}
	$\poly_{bag}(1, 1, 0) = 1 \cdot 1 + 1\cdot 0 + 0 \cdot 1 = 1$
	\end{minipage}\\
\end{tabular}
$$

The Set-PDB query is satisfied in this possible world and the Bag-PDB result tuple has a multiplicity of 1.
The marginal probability (resp., expected count) of this query is computed over all possible worlds:
{\small
\begin{align*}
\probOf[\poly_{set}] &= \hspace*{-1mm}
 \sum_{w_i \in \{\top,\bot\}} \indicator{\poly_{set}(w_a, w_b, w_c)}\probOf[W_a = w_a,W_b = w_b,W_c = w_c]\\
\expct[\poly_{bag}] &= \sum_{w_i \in \{0,1\}} \poly_{bag}(w_a, w_b, w_c)\cdot \probOf[W_a = w_a,W_b = w_b,W_c = w_c]
\end{align*}
}
\end{Example}

Note that the query of \Cref{ex:bag-vs-set} in set semantics is indeed non-hierarchical~\cite{DS12}, and thus \sharpphard.
To see why computing this probability is hard, observe that the three clauses $(W_aW_b, W_bW_c,\text{ and }W_aW_c)$ of $(\ref{eq:poly-set})$ are not independent (the same variables appear in multiple clauses) nor disjoint (the clauses are not mutually exclusive).  Computing the probability of such formulas exactly requires exponential time algorithms (e.g., Shanon Decomposition). 
Conversely, in Bag-PDBs, correlations between monomials of the SOP polynomial (\ref{eq:poly-bag}) are not problematic thanks to linearity of expectation.
The expectation computation over the output lineage is simply the sum of expectations of each clause.
For \Cref{ex:intro}, the expectation is simply
\begin{equation*}
\expct\pbox{\poly_{bag}(W_a, W_b, W_c)} = \expct\pbox{W_aW_b} + \expct\pbox{W_bW_c} + \expct\pbox{W_cW_a}
\end{equation*}
In this particular lineage polynomial, all variables in each product clause are independent, so we can push expectations through.
\begin{equation*}
= \expct\pbox{W_a}\expct\pbox{W_b} + \expct\pbox{W_b}\expct\pbox{W_c} + \expct\pbox{W_c}\expct\pbox{W_a}
\end{equation*}

Computing such expectations is indeed linear in the size of the SOP as the number of operations in the computation is \textit{exactly} the number of multiplication and addition operations of the polynomial.
As a further interesting feature of this example, note that $\expct\pbox{W_i} = \probOf[W_i = 1]$, and so taking the same polynomial over the reals:
\begin{equation}
\label{eqn:can-inline-probabilities-into-polynomial}
\expct\pbox{\poly_{bag}}
= \poly_{bag}(\probOf[W_a=1], \probOf[W_b=1], \probOf[W_c=1])
\end{equation}


\subsection{Superlinearity of Bag PDBs}
Moving forward, we focus exclusively on bags and drop the subscript from $\poly_{bag}$.
Consider the product of $\poly$ with itself:
\begin{equation*}
\poly^2() := \rel(A), E(A, B), \rel(B),\; \rel(C), E(C, D), \rel(D)
\end{equation*}
For an arbitrary polynomial, it is known that there may exist equivalent compressed representations.
One such compression is the factorized polynomial~\cite{factorized-db}, where the polynomial is broken up into separate factors.
For example:
{\small
\begin{equation*}
\poly^2(W_a, W_b, W_c) = \left(W_aW_b + W_bW_c + W_cW_a\right) \cdot \left(W_aW_b + W_bW_c + W_cW_a\right)
\end{equation*}
}
This factorized expression can be easily modeled as an expression tree, as in \Cref{fig:circuit-q2-intro},
 while the equivalent SOP representation is
\begin{equation*}
W_a^2W_b^2 + W_b^2W_c^2 + W_c^2W_a^2 + 2W_a^2W_bW_c + 2W_aW_b^2W_c + 2W_aW_bW_c^2.
\end{equation*}
The expectation $\expct\pbox{\poly^2(W_a, W_b, W_c)}$ then is:
\begin{footnotesize}
\begin{equation*}
\expct\pbox{W_a^2}\expct\pbox{W_b^2} + \expct\pbox{W_b^2}\expct\pbox{W_c^2} + \expct\pbox{W_c^2}\expct\pbox{W_a^2} + 2\expct\pbox{W_a^2}\expct\pbox{W_b}\expct\pbox{W_c} + 2\expct\pbox{W_a}\expct\pbox{W_b^2}\expct\pbox{W_c} + 2\expct\pbox{W_a}\expct\pbox{W_b}\expct\pbox{W_c^2}
\end{equation*}
\end{footnotesize}
Recall the nice property of $\query$ that its expected count could be computed by evaluating its lineage on the probability vector (i.e., \Cref{eqn:can-inline-probabilities-into-polynomial}).
This property does not hold for $\poly^2$ (i.e., $\expct\pbox{\poly^2} \neq \poly^2(\probOf\pbox{W_a}, \probOf\pbox{W_b}, \probOf\pbox{W_c})$), but does suggest a related closed form formula.
Note that if $Dom(W_i) = \{0, 1\}$, then for any $k > 0$, $\expct\pbox{W_i^k} = \expct\pbox{W_i}$.
This property leads us to consider a structure related to $\poly$.
For any polynomial $\poly(\vct{X})$, we define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in $\poly(\vct{X})$ to $1$.
With $\poly^2$ as an example, we have:
\begin{align*}
\rpoly^2(W_a, W_b, W_c)
=&\; W_aW_b + W_bW_c + W_cW_a + 6W_aW_bW_c
\end{align*}
Note that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(\probOf\pbox{W_a=1}, \probOf\pbox{W_b=1}, \probOf\pbox{W_c=1})$).
Also note that the $\poly$ in~\Cref{ex:bag-vs-set} is already in reduced form.

The reduced form of a polynomial can be obtained in a linear scan over the clauses of an SOP encoding of the polynomial.
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 general however, compressed encodings of the polynomial can be exponentially smaller in $k$ for $k$-products --- the query $\poly^k$ obtained by taking the Cartesian product of $k$ copies of $\poly$ has 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?}
\end{quote}
If so, then Bag-PDBs can compete with deterministic databases.
This is ufortunately not the case, and an approximation is required.

\subsection{Overview of our results and techniques}
Concretely, in this paper:
(i) We show that conjunctive queries over a bag-$\ti$ are hard (i.e., superlinear in the size of a compressed lineage encoding) by reduction from counting the number of $k$-matchings over an arbitrary graph;
(ii) We present an $(1-\epsilon)$-approximation algorithm for bag-$\ti$s and show that its complexity is linear in the size of the compressed lineage encoding;
(iii) We generalize the approximation algorithm to bag-$\bi$s, a more general model of probabilistic data;
(iv) We further generalize our results to higher moments, polynomial circuits, and prove that for RA+ queries, the processing time in approximation is within a constant factor of the same query processed deterministically.

Our hardness results follow by considering a suitable generalization of the lineage polynomial in Example~\ref{ex:bag-vs-set}. First it is easy to generalize the polynomial in Example~\ref{ex:bag-vs-set} to $\poly_G(X_1,\dots,X_n)$ that represents the edge set of a graph $G$ in $n$ vertices. Then $\poly_G^k(X_1,\dots,X_n)$ (i.e., $\inparen{\poly_G(X_1,\dots,X_n)}^k$) encodes as its monomials all subgraphs of $G$ with at most $k$ edges in it. This implies that the corresponding reduced polynomial $\rpoly_G^k(\prob,\dots,\prob)$ can be written as $\sum_{i=0}^{2k} c_i\cdot \prob^i$ and we observe that $c_{2k}$ is proportional to the number of $k$-matchings (computing which is \sharpwonehard\ ) in $G$. Thus, if we have access to $\rpoly_G^k(\prob_i,\dots,\prob_i)$ for distinct values of $\prob_i$ for $0\le i\le  2k$, then we can set up a system of linear equations and compute $c_{2k}$ (and hence the number of $k$-matchings in $G$). This result, however, does not rule out the possibility that computing $\rpoly_G^k(\prob,\dots, \prob)$ for a {\em single specific} value of $\prob$ might be easy: indeed  it is easy for $\prob=0$ or $\prob=1$. However, we are able to show that for any other value of $\prob$, computing $\rpoly_G^k(\prob,\dots, \prob)$ exactly will most probably require super-linear time. This reduction needs more work (and we cannot yet extend our results to $k>3$). Further, we have to rely on more recent conjectures in {\em fine-grained} complexity on e.g. the complexity of counting the number of triangles in $G$ and not more standard parameterized hardness like \sharpwonehard.


The starting point of our approximation algorithm was the simple observation that for any lineage polynomial $\poly(X_1,\dots,X_n)$, we have $\rpoly(1,\dots,1)=Q(1,\dots,1)$ and if all the coefficients of $\poly$ are constants, then $\poly(X_1,\dots,X_n)$ (which can be easily computed in linear time) is a $\prob^k$ approximation to the value $\rpoly(\prob,\dots, \prob)$ that we are after. If $\prob$ (i.e., the \emph{input} tuple probabilities) and $k=\deg(\poly)$ are constants, then this gives a constant factor approximation. We then use sampling to get a better approximation factor of $(1\pm \eps)$: we sample monomials from $\poly(1,\dots,1)$ and do an appropriate weighted sum of their coefficients. Standard tail bounds then allow us to get our desired approximation scheme. To get a linear runtime, it turns out that we need the following properties from our compressed representation of $\poly$: (i) be able to compute $\poly(X_1,\dots,X_n)$ in linear time and (ii) be able to sample monomials from $\poly(X_1,\dots,X_n)$ quickly as well. For the ease of exposition, we start off with expression trees (see~\Cref{fig:circuit-q2-intro} for an example) and show that they satisfy both of these properties. Later we show that it is easy to show that these properties also extend to polynomial circuits as well (we essentially show that in the required time bound, we can simulate access to the `unrolled' expression tree by considering the polynomial circuit).

We also formalize our claim that, since our approximation algorithm runs in time linear in the size of the polynomial circuit, we can approximate the expected output tuple multiplicities with only a $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to higher moments of the tuple multiplicity (instead of just the expectation).

\paragraph{Paper Organization.} We present some relevant background and set up our notation in~\Cref{sec:background}. We present our hardness results in~\Cref{sec:hard} and our approximation algorithm in~\Cref{sec:algo}. We present some (easy) generalizations of our results in~\Cref{sec:gen}. We do a quick overview of related work in~\Cref{sec:related-work} and conclude with some open questions in~\Cref{sec:concl-future-work}.

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