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.