diff --git a/class_33_07112023/summary.txt b/class_33_07112023/summary.txt
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+Undecidability of validity problem in FOL  
+
+PCP - examples 
+PCP to validity reduction
+
+Let x and y be the strings over {0,1}. 
+Let K be an instance of PCP.
+That is ((x_1,y_1),(x_2,y_2).....(x_k,y_k))
+where for all  i \in {1,..,k} x_i, y_i are strings over {0,1}. 
+Then the set of indices i_1,i_2,..i_n \in {1,..,k} is a solution if 
+x_{i_1}x_{i_2}...x_{i_n}= y_{i_1}y_{i_2}...y_{i_n}. 
+
+
+From instance  k construct \phi_k.
+
+
diff --git a/class_34_07112023/summary.txt b/class_34_07112023/summary.txt
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+Correctness of PCP to validity problem reduction
+
+K has a solution iff \phi_k is valid. 
+
diff --git a/class_35_09112023/summary.txt b/class_35_09112023/summary.txt
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+midsem answersheets
+