GATE | GATE-CS-2017 (Set 2) | Question 50

Let L(R) be the language represented by regular expression R. Let L(G) be the language generated by a context free grammar G. Let L(M) be the language accepted by a Turing machine M. Which of the following decision problems are undecidable?

I. Given a regular expression R and a string w, is w∈L(R)?

II. Given a context-free grammar G, is L(G)=∅

III. Given a context-free grammar G, is L(G)=Σ^∗ for some alphabet Σ?

IV. Given a Turing machine M and a string w, is w ∈ L(M)?
(A) I and IV only
(B) II and III only
(C) II, III and IV only
(D) III and IV only


Answer: (D)

Explanation:

1. Membership problem of regular language ->Decidable
2. Emptyness of CFL ->Decidable
3. L(G)=Σ^∗ problem of CFL  -> UNdecidable
4. Membership problem of RE language -> UNdecidable

Therefore , option D 

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