# Various Properties of context free languages (CFL)

Context-Free Language (CFL) is a language which is generated by a context-free grammar or Type 2 grammar(according to Chomsky classification) and gets accepted by a Pushdown Automata.

Some very much important properties of a context-free language is:

Regularity- context-free languages are Non-Regular PDA language.

**Closure properties**** :**

The context-free languages are closed under some specific operation, closed means after doing that operation on a context-free language the resultant language will also be a context-free language.Some such operation are:

- Union Operation
- Concatenation
- Kleene closure
- Reversal operation
- Homomorphism
- Inverse Homomorphism
- Substitution
- init or prefix operation
- Quotient with regular language
- Cycle operation
- Union with regular language
- Intersection with regular language
- Difference with regular language

Context free language is not closed under some specific operation, not-closed means after doing that operation on a context-free language the resultant language not remains be a context-free language anymore.

Some such operation are:

- Intersection
- Complement
- Subset
- Superset
- Infinite Union
- Difference, Symmetric difference (xor, Nand, nor or any other operation which get reduced to intersection and complement

- Test for Membership: Decidable.
- Test for Emptiness: Decidable
- Test for finiteness: Decidable

Rest the decision properties are undecidable in context-free language.

**Deterministic property :**

The context-free language can be:

- DCFL-Deterministic (which can be recognized by deterministic pushdown automata) context-free language
- NDCFL-Non-deterministic (can’t be recognized by DPDA but NPDA) context free language.