Update reproducing.m, PTR-BCR-4

Volume_1/Book_about_Quadratization.tex

@@ -1593,7 +1593,7 @@ \subsection{PTR-BCR-4 (Boros, Crama, and Rodr\'{i}guez-Heck, 2018)}
 %Suppose we have a monomial $b_1b_2b_3 \ldots b_k$.
 This is a more general form of the previous reduction, PTR-BCR-4, in which $k=2^{m+1}$:
 \begin{align}
-b_1 \ldots b_k &\rightarrow  \sum_{ij}b_ib_j + \sum_{ij}^m 2^{i+j}b_{a_i}b_{a_j} - \sum_i \sum_j^m 2^{j+1}b_ib_{a_i}.
+b_1 \ldots b_k &\rightarrow  \sum_{ij}b_ib_j + \sum_{ij}^m 2^{i+j}b_{a_i}b_{a_j} - \sum_i \sum_j^m 2^{j+1}b_ib_{a_j}.
 \end{align}
 
 \costsec
@@ -1619,7 +1619,12 @@ \subsection{PTR-BCR-4 (Boros, Crama, and Rodr\'{i}guez-Heck, 2018)}
 \begin{eqnarray}
 %b_1 b_2 b_3 b_4 = \min_{b_{a_1}, b_{a_2}} \frac{1}{2}
 b_1 b_2 b_3 b_4 \rightarrow
-\left( b_1 + b_2 + b_3 + b_4 - 2b_a \right)^2
+\left( b_1 + b_2 + b_3 + b_4 - b_{a_1} - 2b_{a_2} \right)^2
+\end{eqnarray}
+
+\begin{eqnarray}
+b_1 b_2 b_3 b_4 \rightarrow
+\sum_{i=1}^{4}\sum_{j=1}^{4}b_ib_j + \sum_{i=0}^{1}\sum_{j=0}^{1} 2^{i+j}b_{a_i}b_{a_j} - \sum_{i=1}^{4}\sum_{j=0}^{1} 2^{j+1}b_ib_{a_j}
 \end{eqnarray}
 
 \altformsec

everything_else/reproducing.m

@@ -157,6 +157,81 @@
 %% Pg. 21, PTR-BCR-2
 %% Pg. 22, PTR-BCR-3 (example appears to be the same as PTR-BCR-1, and may have to be redone)
 %% Pg. 23, PTR-BCR-4 
+
+% Eq. 72
+b = dec2bin(2^6-1:-1:0)-'0';
+b1=b(:,1);b2=b(:,2);b3=b(:,3);b4=b(:,4);ba1=b(:,5);ba2=b(:,6);
+LHS = min(reshape(b1.*b2.*b3.*b4, 4, []));
+RHS = min(reshape((b1 + b2 + b3 + b4 - ba1 - 2*ba2).^2, 4, []));
+isequal(LHS,RHS); % Gives 1, confirmed by Nike on 6 April.
+
+% Eq. 73
+b = dec2bin(2^6-1:-1:0)-'0';
+LHS = ones(2^6,1);
+for i = 1:4
+    LHS = LHS.*b(:,i);
+end
+LHS = min(reshape(LHS, 4, []));
+RHS = zeros(2^6,1);
+for i = 1:4
+    for j = 1:4
+        RHS = RHS + b(:,i).*b(:,j);
+    end
+end
+for i = 5:6
+    for j = 5:6
+        RHS = RHS + 2^(i+j-10)*b(:,i).*b(:,j);
+    end
+end
+for i = 1:4
+    for j = 5:6
+        RHS = RHS - 2^(j-4)*b(:,i).*b(:,j);
+    end
+end
+RHS = min(reshape(RHS, 4, []));
+isequal(LHS, RHS);
+
+% k = 8, Eq. 74
+b = dec2bin(2^11-1:-1:0)-'0';
+LHS = ones(2^11,1);
+for i = 1:8
+    LHS = LHS.*b(:,i);
+end
+LHS = min(reshape(LHS, 8, []));
+RHS = zeros(2^11,1);
+for i=1:8
+    RHS = RHS + b(:,i);
+end
+RHS = RHS - b(:,9) - 2*b(:,10) - 4*b(:,11);
+RHS = min(reshape(RHS.^2, 8, []));
+isequal(LHS, RHS);
+
+% k = 8, Eq. 71
+b = dec2bin(2^11-1:-1:0)-'0';
+LHS = ones(2^11,1);
+for i = 1:8
+    LHS = LHS.*b(:,i);
+end
+LHS = min(reshape(LHS, 8, []));
+RHS = zeros(2^11,1);
+for i = 1:8
+    for j = 1:8
+        RHS = RHS + b(:,i).*b(:,j);
+    end
+end
+for i = 9:11
+    for j = 9:11
+        RHS = RHS + 2^(i+j-18)*b(:,i).*b(:,j);
+    end
+end
+for i = 1:8
+    for j = 9:11
+        RHS = RHS - 2^(j-8)*b(:,i).*b(:,j);
+    end
+end
+RHS = min(reshape(RHS, 8, []));
+isequal(LHS, RHS);
+
 %% Pg. 24, PTR-KZ (needs an example!)
 %% PTR-KZ: b1b2b3 = min_ba(1 − (ba + b1 + b2 + b3) + ba (b1 + b2 + b3) + b1b2 + b1b3 + b2b3)