\subsection{Superlinearity of Bag PDBs}\label{sec:suplin-bags}
Moving forward, we focus exclusively on bags. For $Q():-$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$$OnTime(\text{City}')$ over the bag relations of \cref{fig:ex-shipping-simp}, consider the product query $\poly^2():- Q \times Q$.
The factorized representation of $\poly^2$ is (for simplicity we ignore the random variables of $Route$ since each variable has probability of $1$):
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 $\poly$.
\begin{Definition}\label{def:reduced-poly}
For any polynomial $\poly(\vct{X})$, define the \emph{reduced polynomial}$\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in $\poly(\vct{X})$ to $1$.
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 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?}
%\end{quote}
%If so, then Bag-PDBs can indeed compete with deterministic databases.
%This is unfortunately not the case, and an approximation is required.
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 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: $\query(\idb)=\comprehension{\query(\db)}{\db\in\idb}$
For a probabilistic database $\pdb=(\idb, \pd)$, the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$ that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
Note that in this work, for the query output, we consider bags, i.e., each possible world in the query output is a set of bag relations and queries are evaluated using bag semantics. We will use $\domK$-relations to model bags. A \emph{$\domK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK=(\domK, \addK, \multK, \zeroK, \oneK)$. 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 $\multK$.
Formally, an n-ary $\semK$-relation 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 a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.
We review positive relational algebra semantics for $\semK$-relations below.
Consider the semiring $\semN=(\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb=(\idb, \pd)$, query $\query$, and possible result tuple $t$, we use $\query(\db)(t)$ for $\db\in\idb$ as input in RHS of \cref{eq:intro-bag-expectation} to compute the expected multiplicity of $t$. Note that the tables of \cref{fig:ex-shipping-simp} have an implicit $1$$\semN$-valued annotation for each tuple in tables $OnTime$ and $Route$.
%\cref{ex:intro-tbls} and \cref{ex:intro-lineage} $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \probDist}[\query(\db)(t)]$:
We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$. Below, we assume that tuples are of appropriate arity, use $\sch(\rel)$ to denote the attributes of $\rel$, and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$. Furthermore, $\theta(\tup)$ denotes the (Boolean) result of evaluating condition $\theta$ over $\tup$.
Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$, with the standard addition and multiplication of polynomials.
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 an equivalent $\semN$-PDB) are both formalized in \Cref{subsec:supp-mat-background}). \AR{Boris/Oliver: Should we mention that the proposition is obvious but has not been stated in literature for bags?}
This proposition shows that computing expected tuple multiplicities is equivalent to computing the expectation of a polynomial (for that tuple) from a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$.
We focus on this problem from now on, assume an implicit result tuple, and so drop the subscript from $\polyForTuple$ (i.e., $\poly$ will denote a polynomial).
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 $\pxdb(\tup)\neq0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same blocks are disjoint events.