We know that Context Free Languages (CFLs) are not closed under intersection operation. However, if L is a context free language, and R is a regular language, L R is context free. We show that using a specific example.

1) Design a DFA for language R = { w | |w|%3 = 1} where % is the modulo operation. That is the length of strings is either 1, 4, 7, . . ..

2) Consider the PDA for the context free language L = {0 n1 n | n > 0}. Construct a PDA for the the language L R

3) Explain the intuition behind your construction, i.e., explain the reason for all the states in DFA and the transitions/operations with the stack. Your construction method should be generalizable for any given DFA and PDA and should not rely on the specific properties of L and R.