Finished pass on Section 4 (Aaron)

approx_alg.tex

@@ -45,19 +45,18 @@ $
 
 \end{Definition}
 \revision{
-Note that similar in spirit to \Cref{def:reduced-bi-poly}, $\expansion{\circuit}$ reduces all variable exponents $e > 1$ to $e = 1$, though \Cref{def:reduced-bi-poly} is more general.
-\OK{More general, how?}
+Note that similar in spirit to \Cref{def:reduced-bi-poly}, $\expansion{\circuit}$ reduces all variable exponents $e > 1$ to $e = 1$.
 }
 
 In the following, we abuse notation and write $\monom$ to denote the monomial obtained as the products of the variables in the set.
 
 \begin{Example}\label{example:expr-tree-T}
 Consider the factorized representation $(X+ 2Y)(2X - Y)$ of the polynomial in~\Cref{eq:poly-eg}.
-Its circuit $\etree$ is illustrated in \cref{fig:circuit}.
+Its circuit $\circuit$ is illustrated in \cref{fig:circuit}.
 The pure expansion of the product is $2X^2 - XY + 4XY - 2Y^2$ and the $\expansion{\circuit}$ is $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
 \end{Example}
-$\expansion{\circuit}$ encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
-However, unlike $\rpoly$, $\expansion{\circuit}$ does not need to be in SOP form.
+$\expansion{\circuit}$ effectively\footnote{The minor difference here is that $\expansion{\circuit}$ encodes the \emph{reduced} form over the SOP expansion of the compressed representation, as opposed to the \abbrSMB representation} encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
+However, unlike the constraint on the input to compute $\rpoly$, the input circuit $\circuit$ does not need to be in \abbrSMB/SOP form.
 
 \begin{Definition}[Positive \circuit]\label{def:positive-circuit}
 For any circuit $\circuit$, the corresponding
@@ -83,11 +82,11 @@ The function \depth~ has circuit $\circuit$ as input and outputs the number of l
 
 \begin{Definition}[$\degree(\cdot)$]
 \revision{
-$\deg(\circuit)$ is defined recursively as follows:
-\[\deg(\circuit)=
+$\degree(\circuit)$ is defined recursively as follows:
+\[\degree(\circuit)=
 \begin{cases}
-\max(\deg(\circuit_\linput),\deg(\circuit_\rinput)) & \text{ if }\circuit.type=+\\
-\deg(\circuit_\linput) + \deg(\circuit_\rinput)+1 &\text{ if }\circuit.type=\times\\
+\max(\degree(\circuit_\linput),\degree(\circuit_\rinput)) & \text{ if }\circuit.type=+\\
+\degree(\circuit_\linput) + \degree(\circuit_\rinput)+1 &\text{ if }\circuit.type=\times\\
 0 & \text{otherwise}.
 \end{cases}
 \]
@@ -214,16 +213,14 @@ we first state the lemmas that summarize the relevant properties of $\onepass$ a
 \begin{Lemma}\label{lem:one-pass}
 The $\onepass$ function completes in time:
 $$O\left(size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
-%, where $N = \size(\circuit)$.\footnote{In the appendix we give a sufficient condition when $\abs{\circuit}(1,\ldots, 1)$ is indeed $O(1)$ in arithmetic computations.  Most notably, WCOJ and FAQ results are not affected by the general runtime of arithmetic computations, a point which we also address in the appendix.}
   $\onepass$ guarantees two post-conditions:  First, for each subcircuit $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$.  Second, when $\vari{S}.\type  = \circplus$, \subcircuit.\lwght $= \frac{\abs{\subcircuit_\linput}(1,\ldots, 1)}{\abs{\subcircuit}(1,\ldots, 1)}$ and likewise for \subcircuit.\rwght.
 \end{Lemma}
-To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}$, $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
+To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
 
 \begin{Lemma}\label{lem:sample}
 The function $\sampmon$ completes in time
 $$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
-%\footnote{Note that the same sufficient condition on \circuit to guarentee $O(1)$ arithmetic computations applies here, and when this condition is met, the runtime loses the $\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}}$ factor},
- where $k = \degree(\circuit)$.  The function returns every $\left(\monom, sign(\coef)\right)\in \expansion{\abs{\circuit}}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$. 
+ where $k = \degree(\circuit)$.  The function returns every $\left(\monom, sign(\coef)\right)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$. 
 \end{Lemma}
 
 With the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
@@ -258,7 +255,7 @@ It turns out that for proof of~\Cref{lem:sample}, we need to argue that when $\c
 \circuit.\rwght &\gets \frac{\abs{\circuit_\rinput}(1,\ldots, 1)}{\abs{\circuit_\linput}(1,\ldots, 1)+ \abs{\circuit_\rinput}(1,\ldots, 1)}
 \end{align}
 
-\noindent \onepass\ (Algorithm ~\ref{alg:one-pass-iter} in \Cref{sec:proofs-approx-alg}) iteratively visits each gate one time according to the topological ordering of \circuit annotating the \lwght, \rwght, and \prt variables of each node according to the definitions above.  Lemma~\ref{lem:one-pass} is also proved in~\Cref{sec:proofs-approx-alg}.
+\noindent \onepass\ (Algorithm ~\ref{alg:one-pass-iter} in \Cref{sec:proofs-approx-alg}) iteratively visits each gate one time according to the topological ordering of \circuit annotating the \lwght, \rwght, and \prt variables of each node according to the definitions above.  Lemma~\ref{lem:one-pass} is proved in~\Cref{sec:proofs-approx-alg}.
 
 \subsection{\sampmon\ Algorithm}
 \label{sec:samplemonomial}

mult_distinct_p.tex

@@ -2,8 +2,7 @@
 %!TEX root=./main.tex
 \section{Hardness of exact computation}
 \label{sec:hard}
-\AH{The notation used here is different than in~\Cref{sec:background}, in particular~\Cref{eq:expect-q-nx}.  Maybe we should  decide on a notation and try to stick to it as much as possible?}
-\BG{We sometimes use $\expct_{\vct{X} \sim P}$ sometimes $\expct_{\vct{X}}$}
+
 In this section, we will prove that computing $\expct\limits_{\vct{W} \sim \pd}\pbox{\poly(\vct{W})}$ exactly for a \ti-lineage polynomial  $\poly(\vct{X})$ generated from a project-join query is \sharpwonehard. Note that this implies hardness for \bis and general $\semNX$-PDBs. Furthermore, we demonstrate in \Cref{sec:single-p} that the problem remains hard, even if $\probOf(X_i) = \prob$ for all $X_i$ and any fixed valued $\prob \in (0, 1)$ as long as certain popular hardness conjectures in fine-grained complexity hold.