CFL Closure Property

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Context-free languages are closed under −

• Union
• Concatenation
• Kleene Star operation

Union

Let L₁ and L₂ be two context free languages. Then L₁ ∪ L₂ is also context free.

Example

Let L₁ = { aⁿbⁿ , n > 0}. Corresponding grammar G₁ will have P: S1 → aAb|ab

Let L₂ = { c^md^m , m ≥ 0}. Corresponding grammar G₂ will have P: S2 → cBb| ε

Union of L₁ and L₂, L = L₁ ∪ L₂ = { aⁿbⁿ } ∪ { c^md^m }

The corresponding grammar G will have the additional production S → S1 | S2

Concatenation

If L₁ and L₂ are context free languages, then L₁L₂ is also context free.

Example

Union of the languages L₁ and L₂, L = L₁L₂ = { aⁿbⁿc^md^m }

The corresponding grammar G will have the additional production S → S1 S2

Kleene Star

If L is a context free language, then L* is also context free.

Example

Let L = { aⁿbⁿ , n ≥ 0}. Corresponding grammar G will have P: S → aAb| ε

Kleene Star L₁ = { aⁿbⁿ }*

The corresponding grammar G₁ will have additional productions S1 → SS₁ | ε

Context-free languages are not closed under −

• Intersection − If L1 and L2 are context free languages, then L1 ∩ L2 is not necessarily context free.

• Intersection with Regular Language − If L1 is a regular language and L2 is a context free language, then L1 ∩ L2 is a context free language.

• Complement − If L1 is a context free language, then L1’ may not be context free.