GATE CS 2021 Set 1 — Question 22
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Theory of Computation → Context-Free Languages → Decision Problems (CFL)
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Question
Let denote an encoding of an automaton . Suppose that . Which of the following languages is/are NOT recursive?
Correct answer
(D) L = \ M M is a PDA such that L(M) = Σ^ \
Solution
We analyze the decidability (recursiveness) of each language:
(A) Emptiness of a DFA is decidable. We can check reachability from the start state to any final state. Thus, is recursive.
(B) Universality of a DFA () is decidable. We can construct the complement DFA and check if . Thus, is recursive.
(C) Emptiness of a PDA is decidable. We can convert the PDA to a Context-Free Grammar (CFG) and check if the start symbol generates any terminal string. Thus, is recursive.
(D) Universality of a PDA () is undecidable. The problem of determining whether a CFG generates all strings is undecidable. Thus, is NOT recursive.The question asks for languages that are NOT recursive.
(A) Emptiness of a DFA is decidable. We can check reachability from the start state to any final state. Thus, is recursive.
(B) Universality of a DFA () is decidable. We can construct the complement DFA and check if . Thus, is recursive.
(C) Emptiness of a PDA is decidable. We can convert the PDA to a Context-Free Grammar (CFG) and check if the start symbol generates any terminal string. Thus, is recursive.
(D) Universality of a PDA () is undecidable. The problem of determining whether a CFG generates all strings is undecidable. Thus, is NOT recursive.The question asks for languages that are NOT recursive.
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