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Minor changes to def 2.7, def 2.9 and the prose between them.

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Aaron Huber 2022-06-01 11:10:09 -04:00
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1 changed files with 8 additions and 7 deletions
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prob-def.tex
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@ -11,9 +11,10 @@ We focus on the problem of computing $\expct_{\worldvec\sim\pdassign}\pbox{\apol
\begin{Definition}[Circuit]\label{def:circuit}
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) with source gates (in degree of $0$) drawn from either $\domN$ or $\vct{X} = \inparen{X_1,\ldots,X_\numvar}$ and one sink gate for each result tuple. Internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
%
Each gate has the following members: \type, \vari{input}, \val, \vpartial, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an extra member \val for the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input of \circuit.
Each gate has the following members: \type, \vari{input}, %\val,
\vpartial, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an extra member \val for the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input of \circuit.
\end{Definition}
We refer to the structure when the underlying DAG is a tree (with edges pointing towards the root) as an expression tree \etree. %In such a case, the root of \etree is analogous to the sink of \circuit. The fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} are used in the proofs of \Cref{sec:proofs-approx-alg}.
We refer to the structure when the underlying DAG is a tree (with edges pointing towards the root) as an expression tree \etree. Members not described in the definition are defined and used in the appendix proofs. %In such a case, the root of \etree is analogous to the sink of \circuit. The fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} are used in the proofs of \Cref{sec:proofs-approx-alg}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The circuits $\inparen{1}$ and $\inparen{2}$ in column $\poly$ of \Cref{fig:two-step} are both expression trees.%encode their respective polynomials in column $\poly$.
%Note that the ciricuit \circuit representing $AX$ and the circuit \circuit' representing $B\inparen{Y+Z}$ each encode a tree, with edges pointing towards the root.
@ -61,7 +62,7 @@ The circuits $\inparen{1}$ and $\inparen{2}$ in column $\poly$ of \Cref{fig:two-
\end{figure}
We next define the inverse of the function $\polyf(\cdot)$ (\Cref{def:poly-func}), which maps a circuit to the polynomial it encodes.
The function $\polyf\inparen{\cdot}$ (\Cref{def:poly-func}) maps a circuit to its corresponding polynomial. We next define the inverse of the function $\polyf(\cdot)$.% (\Cref{def:poly-func}).%, which maps a circuit to the polynomial it encodes.
\begin{Definition}[Circuit Set]\label{def:circuit-set}
$\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $\polyf(\circuit) = \polyX$.
@ -69,7 +70,7 @@ $\circuitset{\polyX}$ is the set of all possible circuits $\circuit$ such that $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. One can think of $\circuitset{\polyX}$ as the infinite set of circuits where for each element \circuit, $\polyf\inparen{\circuit} = \polyX$.
The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$. %One can think of $\circuitset{\polyX}$ as the infinite set of circuits where for each element \circuit, $\polyf\inparen{\circuit} = \polyX$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
@ -77,11 +78,11 @@ The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[The Expected Result Multiplicity Problem]\label{def:the-expected-multipl}
Let $\pdb'$ be an arbitrary \abbrCTIDB and $\vct{X}$ be the set of variables annotating tuples in $\tupset'$. Fix an $\raPlus$ query $\query$ and a result tuple $\tup$.
Let $\pdb$ be an arbitrary \abbrCTIDB and $\vct{X}$ be the set of variables annotating tuples in $\tupset$. Fix an $\raPlus$ query $\query$ and a result tuple $\tup$.
The \expectProblem is defined as follows:%\\[-7mm]
\begin{flalign*}
&\textbf{Input}: \circuit \in \circuitset{\polyX} \text{ for }\poly'\inparen{\vct{X}} = \poly'\pbox{\query,\tupset',\tup}&\\
&\textbf{Output}: \expct_{\vct{W} \sim \bpd}\pbox{\poly'\pbox{\query, \tupset', \tup}\inparen{\vct{W}}}.&
&\textbf{Input}: \circuit \in \circuitset{\polyX} \text{ for }\poly\inparen{\vct{X}} = \poly\pbox{\query,\tupset,\tup}&\\
&\textbf{Output}: \expct_{\vct{W} \sim \bpd}\pbox{\poly\pbox{\query, \tupset, \tup}\inparen{\vct{W}}}.&
\end{flalign*}
\end{Definition}