%We formalize our claim from \Cref{sec:intro} that a linear approximation algorithm for our problem implies that PDB queries (under bag semantics) can be answered (approximately) in the same runtime as deterministic queries under reasonable assumptions.
%Lastly, we generalize our result for expectation to other moments.
%\subsection{Cost Model, Query Plans, and Runtime}
%As in the introduction, we could consider polynomials to be represented as an expression tree.
%However, they do not capture many of the compressed polynomial representations that we can get from query processing algorithms on bags, including the recent work on worst-case optimal join algorithms~\cite{ngo-survey,skew}, factorized databases~\cite{factorized-db}, and FAQ~\cite{DBLP:conf/pods/KhamisNR16}. Intuitively, the main reason is that an expression tree does not allow for `sharing' of intermediate results, which is crucial for these algorithms (and other query processing methods as well).
%So far our analysis of \Cref{prob:intro-stmt} has been in terms of the size of the lineage circuits.
%We now show that this model corresponds to the behavior of a deterministic database by proving that for any \raPlus query $\query$, we can construct a compressed circuit for $\poly$ and \bi $\pdb$ of size and runtime linear in that of a general class of query processing algorithms for the same query $\query$ on $\pdb$'s \dbbaseName $\dbbase$.
% Note that by definition, there exists a linear relationship between input sizes $|\pxdb|$ and $|\dbbase|$ (i.e., $\exists c, \db \in \pxdb$ s.t. $\abs{\pxdb} \leq c \cdot \abs{\db})$).
% \footnote{This is a reasonable assumption because each block of a \bi represents entities with uncertain attributes.
% In practice there is often a limited number of alternatives for each block (e.g., which of five conflicting data sources to trust). Note that all \tis trivially fulfill this condition (i.e., $c = 1$).}
%That is for \bis that fulfill this restriction approximating the expectation of results of SPJU queries is only has a constant factor overhead over deterministic query processing (using one of the algorithms for which we prove the claim).
We adopt a minimalistic compute-bound model of query evaluation drawn from the worst-case optimal join literature~\cite{skew,ngo-survey} to define $\qruntime{\cdot,\cdot}$.\AR{Recursive definition needs to change based on what Oliver needs. Also I think in the definition betlow would be better to replace all $\dbbase$ with $D$.}
We assume that full table scans are used for every base relation access. We can model index scans by treating an index scan query $\sigma_\theta(R)$ as a base relation.
It can be verified that worst-case optimal join algorithms~\cite{skew,ngo-survey}, as well as query evaluation via factorized databases~\cite{factorized-db}
%\AR{See my comment on element on whether we should include this ref or not.}
(and work on FAQs~\cite{DBLP:conf/pods/KhamisNR16}) can be modeled as $\raPlus$ queries (though the size of these queries is data dependent).\footnote{This claim can be verified by e.g. simply looking at the {\em Generic-Join} algorithm in~\cite{skew} and {\em factorize} algorithm in~\cite{factorized-db}.} It can be verified that the above cost model on the corresponding $\raPlus$ join queries correctly captures their runtime.
More specifically \Cref{lem:circ-model-runtime} and \Cref{to-be-decided} show that for any $\raPlus$ query $\query$ and $\dbbase$, there exists a circuit $\circuit$ such that $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ and $|\circuit$ are both $O(\qruntime{Q, \dbbase})$. Recall we assumed these two bounds when we moved from \Cref{prob:big-o-joint-steps} to \Cref{prob:intro-stmt}.
% Given an $\raPlus$ query $\query$ over a \ti $\pdb$ with \dbbaseName $\dbbase$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pdb)$ with probability at least $1-\delta$ in time
