Explain the context free language closure under union operation?

If L1 and L2 are CFLs, then their union L1 + L2 is a CFL.Here CFL refers to Context Free Language.

L1 CFL implies that L1 has a CFG, let it is CFG1, that generates it.

• Assume that the nonterminals in CFG1 are S, A, B, C, . . ..

• Change the nonterminal in CFG1 to S1, A1, B1, C1, . . ..

• Don’t change the terminals in the CFG1.

L2 CFL implies that L2 has a CFG, Let it is CFG2, that generates it.

• Assume that the nonterminals in CFG2 are S, A, B, C, . . ..

• Change the nonterminal in CFG2 to S2, A2, B2, C2, . . ..

• Don’t change the terminals in the CFG2.

Now CFG1 and CFG2 have non intersecting sets of nonterminals.

We create a CFG for L1 + L2 as follows −

• Include all of the nonterminal S1, A1, B1, C1, . . . and S2, A2, B2, C2, . . ..

• Include all the productions of CFG1 and CFG2.

Create a new nonterminal S and a production.

S → S1 | S2

Example

CFG1 for L1
S → S | Aa | Bb | ∧
A → Sa | bB | ab
B → S | a
CFG2 for L2
S → aS | aA | Bb | ∧
A → aS |bB| ab
B → Ba | b

To construct CFG for L1 + L2, follow the below mentioned steps −

• Transform CFG1 as follows −

S1 → S1| A1a | B1b | ∧
A1 → S1a | bB1 | ab
B1 → S1 | a

• Transform CFG2 as follows

S2 → aS2 | aA2 | B2b | ∧
A2 → aS2 |bB2| ab
B2 → B2a | b

• Construct CFG for L1 + L2 as follows −

S → S1 | S2
S1 → S1| A1a | B1b | ∧
A1 → S1a | bB1 | ab
B1 → S1 | a
S2 → aS2 | aA2 | B2b | ∧
A2 → aS2 |bB2| ab
B2 → B2a | b