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This is a question from GATE CSE 2009.

Which of the following is FALSE?

A] There is a unique minimal DFA for every regular language.

B] Every NFA can be converted to an equivalent PDA.

C] Complement of every context-free language is recursive.

D] Every non-deterministic PDA can be converted to an equivalent deterministic PDA.

The answer provided to me (without any explanation) is B, which I think is wrong.

Here is my approach.

A] Every RL has an equivalent DFA and there is one unique minimal DFA for a given RL. [so True]

B] Since RL are proper sub-set of CFL and every RL has equivalent FA and every CFL has equivalent PDA, every FA can be converted to PDA but not vice versa. [so True]

C] Given a CFL we can create an equivalent Total TM [RECURSIVE], RECURSIVE languages are closed under complement. [so True]

D] This is only remaining option and by method of elimination answer.

Is my answer correct?

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A is true, but I don't find your argument convincing (it simply restates the claim). See this question for example.

B is not the correct answer to this problem, because as you say it's true. Your reasoning isn't very formal though, and swaps some reduction orders. A correct reduction would be NFA → (superset construction)→ DFA → (trivial) → PDA.

C seems fine to me.

D is indeed left over, and is indeed false. Inherently ambiguous context-free languages can be recognized by nondeterministic PDAs, but not by deterministic PDAs.

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