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Which of the following statements is true?

1. If a language is context free it can always be accepted by a deterministic push-down automaton
2. The union of two context free languages is context free
3. The intersection of two context free languages is a context free
4. The complement of a context free language is a context free

(A) is wrong as a language can be context free even if it is being accepted by non-deterministic PDA for ex- $\{ ww^r: w \in \Sigma^*$ and $w^r$  is reverse of $w\}$

(C) and (D) not always true as Context free languages are not closed under Complement or Intersection.

edited

I think example of A is wrong ... it will be { WWr  :  W ∈(a,b)and W is reverse} ... correct me if i am wrong ...

Edit : { WWr  :  W ∈(a,b)and W is reverse}  ... both r npda .... My mistake ...

Answer is only (B) because Deterministic push down automata can recognize only deterministic context free languages and cannot recognize non- deterministic context free languages.

Non Deterministic push down automata can recognize all the context free languages(both deterministic and non- deterministic).

You can also refer similar GATE question https://gateoverflow.in/3650/gate2004-it-9?show=4336#a4336.

The statement in your answer should have been this - "(A) is wrong as a language can be context-free even if it is being not accepted by deterministic PDA for e.g., ...". We don't care whether a CFL is accepted by a non-deterministic PDA or not. Here we concentrate on Deterministic PDA. The key concept here is NDPA ≠ DPDA, i.e., to say NDPA is more powerful than DPDA.

Ans: B not closed under intersection and complement
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