# Closure Properties Of Context Free Grammar

Context free grammer (CFG) generates the Context free languages (CFL) which are accepted by pushdown automata (PDA) but not by finite automata (FA). Let’s discuss some of the **closure properties of Context free grammer.**

**Closure Properties of CFG**

Consider L1 and If L2 are two context-free languages where L1 = { a^{n}b^{n} | n >= 0 } and L2 = { c^{m}d^{m} | m >= 0 }

**1. CFL is closed under Union**

**As **L3 = L1 ∪ L2 = { a^{n}b^{n} ∪ c^{m}d^{m} | n >= 0, m >= 0 } is also context free.

**2. CFL is closed under Concatenation**

**As **L3 = L1.L2 = { a^{n}b^{n}c^{m}d^{m} | n >= 0, m >= 0 } is also context free.

**3. CFL is closed under Kleen Closure**

L1* = { a^{n}b^{n} | n >= 0 }* is also context free.

**4. CFL is not closed under Complementation**

Complementation of context-free language L1, which is ∑* – L1, is not a CFL.

**5. CFL is not closed under Intersection**

Suppose L1 = { a^{n}b^{n}c^{m} | n >= 0 and m >= 0 } and L2 = (a^{m}b^{n}c^{n} | n >= 0 and m >= 0 } are CFL’s.

Now apply intersection on given CFLs.

L3 = L1 ∩ L2 = { a^{n}b^{n}c^{n} | n >= 0 } is not context-free because there are two comparisons in derived language after intersection.

Note:Deterministic Context-Free Language (DCFL) is closed only under complementation and Inverse Homomorphism