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Moving back circuit complexity related work

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Oliver Kennedy 2020-12-20 01:25:18 -05:00
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related-work-extra.tex
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\section{Compressed Representations of Polynomials and Boolean Formulas}\label{sec:compr-repr-polyn}
There is a large body of work on compact using representations of Boolean formulas (e.g, various types of circuits including OBDDs~\cite{jha-12-pdwm}) and polynomials (e.g.,factorizations~\cite{factorized-db}) some of which have been utilized for probabilistic query processing, e.g.,~\cite{jha-12-pdwm}. Compact representations of Boolean formulas for which probabilities can be computed in linear time include OBDDs, SDDs, d-DNNF, and FBDD. In terms of circuits over semiring expression,~\cite{DM14c} studies circuits for absorptive semirings while~\cite{S18a} studies circuits that include negation (expressed as the monus operation of a semiring). Algebraic Decision Diagrams~\cite{bahar-93-al} (ADDs) generalize BDDs to variables with more than two values. Chen et al.~\cite{chen-10-cswssr} introduced the generalized disjunctive normal form.
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\section{Parameterized Complexity}\label{sec:param-compl}

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related-work.tex
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\section{Related Work}\label{sec:related-work}
In addition to probabilistic databases, our work has connections to work on compact representations of polynomials and on fine-grained complexity, which we review in \Cref{sec:compr-repr-polyn,sec:param-compl}.
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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
Probabilistic Databases (PDBs) have been studied predominantly for set semantics.
\textbf{Probabilistic Databases} (PDBs) have been studied predominantly for set semantics.
A multitude of data models have been proposed for encoding a PDB more compactly than as its set 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, but we use polynomials instead of Boolean expressions and only allow constants as attribute values.
@ -38,6 +35,11 @@ Fink et al.~\cite{FH12} study aggregate queries over a probabilistic version of
% \cite{FH12} identifies a tractable class of queries involving aggregation.
In contrast, we study a less general data model and query class, but provide a linear time approximation algorithm and provide new insights into the complexity of computing expectation (while~\cite{FH12} computes probabilities for individual output annotations).
\textbf{Compressed Encodings} are used extensively for Boolean formulas (e.g, various types of circuits including OBDDs~\cite{jha-12-pdwm}) and polynomials (e.g.,factorizations~\cite{factorized-db}) some of which have been utilized for probabilistic query processing, e.g.,~\cite{jha-12-pdwm}. Compact representations of Boolean formulas for which probabilities can be computed in linear time include OBDDs, SDDs, d-DNNF, and FBDD. In terms of circuits over semiring expression,~\cite{DM14c} studies circuits for absorptive semirings while~\cite{S18a} studies circuits that include negation (expressed as the monus operation of a semiring). Algebraic Decision Diagrams~\cite{bahar-93-al} (ADDs) generalize BDDs to variables with more than two values. Chen et al.~\cite{chen-10-cswssr} introduced the generalized disjunctive normal form.
Additional discussion related work pertaining to fine-grained complexity appears in \Cref{sec:param-compl}.
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