More changes to Def 1.4.

intro-rewrite-070921.tex

@@ -227,7 +227,7 @@ To compute $\expct\pbox{\poly^2}$ we can use linearity of expectation and push t
 
  %the expectation is $\expct\pbox{A^2X^2B^2} = A\cdot\prob_A\cdot\inparen{\sum\limits_{i \in [2]}X_i\cdot \prob_{X, i}}\cdot B\prob_B$ for $X \in \inset{0, 1, 2}$.  
  
- An equivalent representation of $\poly^2$ can be derived by thinking of having a separate product $j\cdot X_j$ for each multiplicity value $j\in\pbox{\bound}$ such that the original `base' variable $X$ is equal to the sum of these products.  For this example, the set of variables could be $\inset{A_1,\ldots,A_4, X_1,\ldots,X_4, B_1,\ldots,B_4}$, where e.g. $X$ now equals $\sum_{j\in\pbox{4}}j\cdot X_j$ and each variable takes values from the set $\inset{0, 1}$.   Our reformulated polynomial $\poly_R^2 = \inparen{\sum_{j_1\in\pbox{\bound}}j_1A_{j_1}}^2$ $\inparen{\sum_{j_2\in\pbox{\bound}}j_2X_{j_2}}^2$ $\inparen{\sum_{j_3\in\pbox{\bound}}j_3B_{j_3}}^2$.  Since tuple multiplicities by nature are disjoint we can drop all cross terms and have $\poly_R^2 = \sum_{j_1, j_2, j_3 \in \pbox{\bound}}j_1^2A^2_{j_1}j_2^2X_{j_2}^2j_3^2B^2_{j_3}$. With the reframed polynomial, the expectation is $\expct\pbox{\poly^2}=\sum_{j_1,j_2,j_3\in\pbox{\bound}}j_1^2j_2^2j_3^2\expct\pbox{A_{j_1}}\expct\pbox{X_{j_2}}\expct\pbox{X_{j_3}}$, since we now have that all $\randWorld_{X_j}\in\inset{0, 1}$.
+ An equivalent representation of $\poly^2$ can be derived by replacing each variable in $\poly^2$ with $\sum_{j\in\pbox{\bound}}j\cdot X_j$, with each $X_j\in\inset{0, 1}$.  Let $\refpoly^2\inparen{A_1,\ldots,A_4, X_1,\ldots,X_4, B_1,\ldots,B_4}$ be the reformulation of polynomial $\poly^2$, where we have substituted the above sum for each $X\in \poly$.  For this example then, $\refpoly^2 = \inparen{\sum_{j_1\in\pbox{\bound}}j_1A_{j_1}}^2$ $\inparen{\sum_{j_2\in\pbox{\bound}}j_2X_{j_2}}^2$ $\inparen{\sum_{j_3\in\pbox{\bound}}j_3B_{j_3}}^2$.  Since the set of multiplicities for tuple $\tup$ by nature are disjoint we can drop all cross terms and have $\poly_R^2 = \sum_{j_1, j_2, j_3 \in \pbox{\bound}}j_1^2A^2_{j_1}j_2^2X_{j_2}^2j_3^2B^2_{j_3}$. Computing  expectation we get $\expct\pbox{\poly^2}=\sum_{j_1,j_2,j_3\in\pbox{\bound}}j_1^2j_2^2j_3^2\expct\pbox{\randWorld_{A_{j_1}}}\expct\pbox{\randWorld_{X_{j_2}}}\expct\pbox{\randWorld_{B_{j_3}}}$, since we now have that all $\randWorld_{X_j}\in\inset{0, 1}$.
 % \begin{footnotesize}
 % \begin{align*}
 % &\expct\pbox{\randWorld_A^2\randWorld_X^2\randWorld_B^2} = \expct\pbox{\randWorld_A^2}\expct\pbox{\inparen{\randWorld_{X_1} + \randWorld_{X_2}}^2}\expct\pbox{\randWorld_B^2} = \expct\pbox{\randWorld_A}\expct\pbox{\randWorld_{X_1}^2 + 2\randWorld_{X_1}\randWorld_{X_2} + \randWorld_{X_2}^2}\expct\pbox{\randWorld_B} =\\
@@ -237,7 +237,7 @@ To compute $\expct\pbox{\poly^2}$ we can use linearity of expectation and push t
 % \end{footnotesize}
 %We can drop the term $\expct\pbox{2\randWorld_{X_1}\randWorld_{X_2}}$ since by definition a tuple can only have one multiplicity value in a possible world, thus always making $\randWorld_{X_1}\cdot \randWorld_{X_2} = 0$.  
 %Another subtlety to note is that for any $i\in \pbox{\bound}$,  $\expct\pbox{\randWorld_{X_i}} = i\cdot\prob_{X, i}$.
-This reformulation of the problem leads us to consider a structure related to the lineage polynomial.
+This leads us to consider a structure related to the lineage polynomial.
 
 %By exploiting linearity of expectation, further pushing expectation through independent variables and observing that for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$, the expectation is
 %$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ (where $\randWorld_A$ is the random variable corresponding to $A$, distributed by $\pdassign$).
@@ -261,11 +261,11 @@ This reformulation of the problem leads us to consider a structure related to th
 %\end{footnotesize}
 %\noindent This property leads us to consider a structure related to the lineage polynomial.
 \begin{Definition}\label{def:reduced-poly}
-For any polynomial $\poly(\vct{X})$ define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by i) replacing all $X_\tup \in \vct{X}$ for $\tup \in \tupset$ with $\sum_{j\in\pbox{\bound}}j\cdot X_{\tup, j}$, i.e. $\rpoly\inparen{\vct{X}}$ has variables $X_{\tup, j}$ for $j \in \pbox{\bound}$ such that $X_{\tup, j} \in \inset{0, 1}$, ii) convert the reformulated polynomial formed into the standard monomial basis (\abbrSMB) 
+For any polynomial $\poly(X_1,\ldots,X_\numvar)$ define the reformulated polynomial $\refpoly\inparen{X_{1, 1},\ldots X_{1, \bound}, X_{2, 1}\ldots X_{\numvar, \bound}}$ to be the polynomial $\refpoly$ = $\poly\inparen{\sum_{j\in\pbox{\bound}}j\cdot X_{1, j},\ldots,\sum_{j\in\pbox{\bound}}j\cdot X_{\numvar, j}}$ and ii) define the \emph{reduced polynomial} $\rpoly(X_{1, 1},\ldots X_{1, \bound}, X_{2, 1}\ldots X_{\numvar, \bound})$ to be the polynomial resulting from converting $\refpoly$ into the standard monomial basis (\abbrSMB) 
 \footnote{
   This is the representation, typically used in set-\abbrPDB\xplural, where the polynomial is reresented as sum of `pure' products. See \Cref{def:smb} for a formal definition.
 }
- while setting all \emph{variable} exponents $e > 1$ to $1$.
+ while setting all \emph{variable} exponents $e > 1$ to $1$ such that $\rpoly = \widetilde{\refpoly}$. 
 \end{Definition}
 Continuing with the example $\poly^2\inparen{A, B, C, E, X_1, X_2, Y, Z}$, to save clutter we i) do not show the full expansion for variables with greatest multiplicity $= 1$ since e.g. for variable $A$, the sum of products itself evaluates to $1^2\cdot A^2 = A$, and ii) for $\sum_{j\in\pbox{\bound}}j^2\cdot X_j$, we omit the summands encoding multiplicities $> 2$, since the greatest multiplicity of the tuple annotated with $X$ is $2$, likewise those summands will always evaluated to $0$ since the tuple will never have a multiplicity of $>2$.  
 \begin{multline*}

macros.tex

@@ -238,6 +238,7 @@
 \newcommand{\atupvar}{\tupvar{\rel}{\tup}}
 \newcommand{\polyX}{\poly\inparen{\vct{\pVar}}}%<---let's see if this proves handy
 \newcommand{\rpoly}{\widetilde{\poly}}%r for reduced as in reduced 'Q'
+\newcommand{\refpoly}{\poly_R}
 \newcommand{\rpolyX}{\rpoly\inparen{\pVar}}%<---if this isn't something we use much, we can get rid of it
 \newcommand{\biDisProd}{\mathcal{B}}%bidb disjoint tuple products (def 2.5)
 \newcommand{\rExp}{\mathcal{T}}%the set of variables to reduce all exponents to 1 via modulus operation; I think \mathcal T collides with the notation used for the set of tuples in D