Update on Overleaf.

binarybidb.tex

@@ -18,7 +18,7 @@ The term $\prod_{\tup\in S} X_\tup^{d_\tup}$ in \Cref{eq:sop-form} is a {\em mon
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Unless othewise noted, we consider all polynomials to be in \abbrSMB representation. 
 When it is unclear, we use $\smbOf{\genpoly}$ 
-to denote the \abbrSMB form of a polynomial $\genpoly~\inparen{\poly}$.
+to denote the \abbrSMB form of a polynomial $\genpoly$.
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %

introduction.tex

@@ -179,14 +179,15 @@ The simple insight to get around this issue to note that the random variables $\
 %\begin{multline*}
 %\refpoly{1, }^{\inparen{ABX}^2}\inparen{A, X, B} =  \poly_1^{\inparen{AXB}^2}\inparen{\sum_{j_1\in\pbox{\bound}}j_1A_{j_1}, \sum_{j_2\in\pbox{\bound}}j_2X_{j_2}, \sum_{j_3\in\pbox{\bound}}j_3B_{j_3}} \\
 %= \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.
-\[\overline{\refpoly{1, }^{\inparen{ABU}^2}}\inparen{A, U_1, U_2 B} =  \poly_1^{\inparen{AUB}^2}\inparen{A,(U_1+2U_2),B}.\]
+\[\refpoly{1, }^{\inparen{ABU}^2}\inparen{A, U_1, U_2 B} =  \poly_1^{\inparen{AUB}^2}\inparen{A,(U_1+2U_2),B}.\]
 %\end{multline*}
 %}
 %Since the set of multiplicities for tuple $\tup$ by nature are disjoint we can drop all cross terms and have $\refpoly{1, }^2 = \sum_{j_1, j_2, j_3 \in \pbox{\bound}}j_1^2A^2_{j_1}j_2^2X_{j_2}^2j_3^2B^2_{j_3}$.  Since we now have that all $\randWorld_{X_j}\in\inset{0, 1}$, computing  expectation yields $\expct\pbox{\refpoly{1, }^2}=\sum_{j_1,j_2,j_3\in\pbox{\bound}}j_1^2j_2^2j_3^2$	\allowbreak $\expct\pbox{\randWorld_{A_{j_1}}}\expct\pbox{\randWorld_{X_{j_2}}}\expct\pbox{\randWorld_{B_{j_3}}}$.
 
-Given that $U$ can only have multiplicity of $1$ or $2$ but not both, we drop the monomials with the term $U_1U_2$ to get 
-$\refpoly{1, }^{\inparen{ABU}^2}\inparen{A, U_1, U_2, B} =  A^2U_1^2B^2+2^2\cdot A^2 U_2^2B^2.$
-Now that world vectors $(\randWorld_A,\randWorld_{U_1},\randWorld_{U_2},\randWorld_A)\in\inset{0,1}^4$, we have $\expct\pbox{\refpoly{1, }^2}=\expct\pbox{\randWorld_{A}}\expct\pbox{\randWorld_{U_1}}\expct\pbox{\randWorld_{B}}+$ \\ $4\expct\pbox{\randWorld_{A}}\expct\pbox{\randWorld_{U_2}}\expct\pbox{\randWorld_{B}}\stackrel{\text{def}}{=}\rpoly_1^2\inparen{p_A,\probOf\inparen{U=1},\probOf\inparen{U=2},p_B}$.  We only did the argument for a single monomial but by linearity of expectation we can apply the same argument to all monomials in $\poly_1^2$. Generalizing this argument to general $\poly$ leads to consider its following `reduced' version:
+Given that $U$ can only have multiplicity of $1$ or $2$ but not both, 
+%we drop the monomials with the term $U_1U_2$ to get 
+%$\refpoly{1, }^{\inparen{ABU}^2}\inparen{A, U_1, U_2, B} =  A^2U_1^2B^2+2^2\cdot A^2 U_2^2B^2.$
+given world vectors $(\randWorld_A,\randWorld_{U_1},\randWorld_{U_2},\randWorld_A)$, we have $\expct\pbox{\randWorld_{U_1}\randWorld_{U_2}}=0$. Further, since the world vectors are Binary vectors, we have $\expct\pbox{\refpoly{1, }^2}=\expct\pbox{\randWorld_{A}}\expct\pbox{\randWorld_{U_1}}\expct\pbox{\randWorld_{B}}+$ \\ $4\expct\pbox{\randWorld_{A}}\expct\pbox{\randWorld_{U_2}}\expct\pbox{\randWorld_{B}}\stackrel{\text{def}}{=}\rpoly_1^2\inparen{p_A,\probOf\inparen{U=1},\probOf\inparen{U=2},p_B}$.  We only did the argument for a single monomial but by linearity of expectation we can apply the same argument to all monomials in $\poly_1^2$. Generalizing this argument to general $\poly$ leads to consider its following `reduced' version:
 
 \begin{Definition}\label{def:reduced-poly}
 For any polynomial $\poly\inparen{\inparen{X_\tup}_{\tup\in\tupset}}$ define the reformulated polynomial $\refpoly{}\inparen{\inparen{X_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}