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GATE | GATE CS 2021 | Set 1 | Question 22
• Last Updated : 24 May, 2021

Let ⟨M⟩ denote an encoding of an automaton M. Suppose that Σ={0,1}. Which of the following languages is/are NOT recursive?
(A) L = {⟨M⟩ ∣ M is a DFA such that L(M)=∅}
(B) L = {⟨M⟩ ∣ M is a DFA such that L(M)=Σ*}
(C) L = {⟨M⟩ ∣ M is a PDA such that L(M)=∅}
(D) L = {⟨M⟩ ∣ M is a PDA such that L(M)=Σ*}

Answer: (D)

Explanation: Recursive language(REC) – A language ‘L’ is said to be recursive if there exists a Turing machine which will accept all the strings in ‘L’ and reject all the strings not in ‘L’. The Turing machine will halt every time and give an answer(accepted or rejected) for each and every string input. A language ‘L’ is decidable if it is a recursive language. All decidable languages are recursive languages and vice-versa.

From the given options, L = {⟨M⟩ ∣ M is a PDA such that L(M)=Σ*} is completeness problem of CFL, which is undecidable. Hence not recursive.

Given a language L, take its complement L’ and check if L’ is empty => L is complete.
Since CFLs are not closed under complementation. So, completeness is undecidable for CFL.

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