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related-work.tex
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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
\textbf{Probabilistic Databases} (PDBs) have been studied predominantly for set semantics.
Many data models have been proposed for encoding PDBs more compactly than as sets of possible worlds.
These include tuple-independent databases~\cite{VS17} (\tis), block-independent databases (\bis)~\cite{RS07}, and \emph{PC-tables}~\cite{GT06} pair a C-table % ~\cite{IL84a}
with probability distribution over its variables.
This is similar to our $\semNX$-PDBs, with Boolean expressions instead of polynomials.
% Tuple-independent databases (\tis) consist of a classical database where each tuple associated with a probability and tuples are treated as independent probabilistic events.
% While unable to encode correlations directly, \tis are popular because any finite probabilistic database can be encoded as a \ti and a set of constraints that ``condition'' the \ti~\cite{VS17}.
% Block-independent databases (\bis) generalize \tis by partitioning the input into blocks of disjoint tuples, where blocks are independent~\cite{RS07}. %,BS06
% \emph{PC-tables}~\cite{GT06} pair a C-table % ~\cite{IL84a}
% with probability distribution over its variables. This is similar to our $\semNX$-PDBs, except that we do not allow for variables as attribute values and instead of local conditions (propositional formulas that may contain comparisons), we associate tuples with polynomials $\semNX$.
These include tuple-independent databases~\cite{VS17} (\tis), block-independent databases (\bis)~\cite{RS07}, and \emph{PC-tables}~\cite{GT06}, which is similar to our $\semNX$-PDBs, with Boolean expressions instead of polynomials.
Approaches for probabilistic query processing (i.e., computing marginal probabilities for tuples), fall into two broad categories.
\emph{Intensional} (or \emph{grounded}) query evaluation computes the \emph{lineage} of a tuple % (a Boolean formula encoding the provenance of the tuple)
\emph{Intensional} (or \emph{grounded}) query evaluation computes the \emph{lineage} of a tuple
and then the probability of the lineage formula.
In this paper we focus on intensional query evaluation with polynomials.
It has been shown that computing the marginal probability of a tuple is \sharpphard~\cite{valiant-79-cenrp} (by reduction from weighted model counting).
The second category, \emph{extensional} query evaluation, % avoids calculating the lineage.
% This approach
is in \ptime, but is limited to certain classes of queries.
Dalvi et al.~\cite{DS12} proved a dichotomy for unions of conjunctive queries (UCQs):
for any UCQ the probabilistic query evaluation problem is either \sharpphard (requires extensional evaluation) or \ptime (permits intensional).
Dalvi et al.~\cite{DS12} proved a dichotomy for UCQs:
for any UCQ the probabilistic query evaluation problem is either \sharpphard or \ptime.
Olteanu et al.~\cite{FO16} presented dichotomies for two classes of queries with negation. % R\'e et al~\cite{RS09b} present a trichotomy for HAVING queries.
Amarilli et al. investigated tractable classes of databases for more complex queries~\cite{AB15}. %,AB15c
Another line of work, studies which structural properties of lineage formulas lead to tractable cases~\cite{kenig-13-nclexpdc,roy-11-f,sen-10-ronfqevpd}.