Explain the Closure Under Kleene Star of CFL in TOC?

If L is a CFL, then L*is a CFL. Here CFL refers to Context Free Language.

Steps

Let CFG for L has nonterminal S, A, B, C, . . ..

Change the nonterminal from S to S1.

We create a new CFG for L* as follows −

• Include all the nonterminal S1, A, B, C, . . . from the CFG for L.

• Include all productions of the CFG for L.

Add new nonterminal S and new production

S → S1S | ∧

We can repeat last production

S → S1S → S1S1S → S1S1S1S → S1S1S1S1S → S1S1S1S1∧ → S1S1S1S1

Note that any word in L* can be generated by the new CFG.

We have to show that any word generated by the new CFG is in L*,

Also, make sure that there is no interaction between the different S1’s.

Example

CFG for L is as follows −

S → PaQ | QQ | ∧
P → SaS | bQb | ab
Q → SS | ba
Convert CFG for L:
S1 → PaQ | QQ | ∧
P → S1aS1 | bQb | ab
Q → S1S1 | ba

New CFG for L* is given below −

S → S1S | ∧
S1 → PaQ | QQ | ∧
P → S1aS1 | bQb | ab
Q → S1S1 | ba