Changed lemma 1.5 and added note to proof regarding using the expanded variables.

app_notation-background.tex

@@ -145,7 +145,7 @@ Finally, note that there are exactly three cases where the expectation of a mono
 \begin{proof}
 Let $\poly$ be a polynomial of $\numvar$ variables with highest degree $= B$, defined as follows: %, in which every possible monomial permutation appears,
 \[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i}.\]
-Let the boolean function $\isInd{\cdot}$ take $\vct{d}$ as input and return true if there does not exist any dependent variables in $\vct{d}$, i.e., $\not\exists ~\block, i\neq j\suchthat d_{\block, i}, d_{\block, j} \geq 1$.\footnote{This \abbrBIDB notation is used and discussed in \cref{subsec:tidbs-and-bidbs}}.
+Note that replacing the variables $X_1,\ldots, X_\numvar$ with $\inset{X_{i, j}~|~ i\in [\numvar], j\in[\bound]}$ and converting to \abbrSMB produces a polynomial that satisfies the above definition.  Let the boolean function $\isInd{\cdot}$ take $\vct{d}$ as input and return true if there does not exist any dependent variables in $\vct{d}$, i.e., $\not\exists ~\block, i\neq j\suchthat d_{\block, i}, d_{\block, j} \geq 1$.\footnote{This \abbrBIDB notation is used and discussed in \cref{subsec:tidbs-and-bidbs}}.
 Then in expectation we have
 \begin{align}
 \expct_{\vct{\randWorld}}\pbox{\poly(\vct{\randWorld})} &= \expct_{\vct{\randWorld}}\pbox{\sum_{\substack{\vct{d} \in \{0,\ldots,B\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i} + \sum_{\substack{\vct{d} \in \{0,\ldots, B\}^\numvar\\\wedge ~\neg\isInd{\vct{d}}}} c_{\vct{d}}\cdot\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar\randWorld_i^{d_i}}\label{p1-s1a}\\

intro-rewrite-070921.tex

@@ -276,10 +276,8 @@ ABX_1 + ABX_2 + BYE + BZC + 2AX_1BYE + 2AX_2BYE + 2AX_1BZC + 2AX_2BZC + 2BYEZC.
 Note that we have argued that for our specific example the expectation that we want is $\widetilde{\poly^2}(\probOf\inparen{A=1},$ $\probOf\inparen{B=1}, \probOf\inparen{C=1}), \probOf\inparen{E=1}, \probOf\inparen{X_1=1}, \probOf\inparen{X_2=1}, \probOf\inparen{Y=1}, \probOf\inparen{Z=1})$.
 %It can be verified that the reduced polynomial parameterized with each variable's respective marginal probability is a closed form of the expected count (i.e., $\expct\limits_{\vct{\randWorld}\sim\pd}\pbox{\Phi^2\inparen{\vct{X}}} = \widetilde{\Phi^2}(\probOf\pbox{A=1},$ $\probOf\pbox{B=1}, \probOf\pbox{C=1}), \probOf\pbox{D=1}, \probOf\pbox{X=1}, \probOf\pbox{Y=1}, \probOf\pbox{Z=1})$).
 \Cref{lem:tidb-reduce-poly} generalizes the equivalence to {\em all} $\raPlus$ queries on \abbrCTIDB\xplural (proof in \Cref{subsec:proof-exp-poly-rpoly}).
-\AH{Here is what is next.  Restate the following lemma in the new setting.}
 \begin{Lemma}\label{lem:tidb-reduce-poly}
-Let $\pdb$ be a $1$-\abbrBIDB such that the probability distribution $\pdassign$ over $\worldvec\in\{0,1\}^{\abs{\tupset}}$ (the set of all worlds) is induced by the disjoint condition and the probability vector $\probAllTup = \inparen{\prob_1,\ldots,\prob_{\abs{\tupset}}}$ where $\prob_i=\probOf\inparen{W_i=1}$.
-For any $1$-\abbrBIDB-lineage polynomial
+For any \abbrCTIDB $\pdb$, $\raPlus$ query.$\query$, and lineage polynomial
 %\BG{Term has not been introduced yet.}
 %Atri: fixed
  $\poly\inparen{\vct{X}}=\apolyqdt(\vct{X})$, it holds that $
@@ -287,6 +285,8 @@ For any $1$-\abbrBIDB-lineage polynomial
 $
 \end{Lemma}
 
+\AH{Here is what I stopped.}
+
 To prove our hardness result we show that for the same $Q$ from the example above, for an arbitrary `product width' $k$, the query $Q^k$ is able to encode various hard graph-counting problems (assuming $\bigO{\numvar}$ tuples rather than the $O(1)$ tuples in \Cref{fig:two-step}).
 We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.