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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
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let L(R) be the language represented by regular expression R. Let L(G)...
- Membership problem of regular language ->Decidable
- Emptyness of CFL ->Decidable
- L(G)=Σ∗ problem of CFL -> UNdecidable
- Membership problem of RE language -> UNdecidable
Therefore , option D
Most Upvoted Answer
Let L(R) be the language represented by regular expression R. Let L(G)...
Undecidable Decision Problems
Explanation:
In the given question, we are asked to determine which of the decision problems are undecidable. To analyze this, let's look at each option one by one.
I. Given a regular expression R and a string w, is w ∈ L(R)?
This problem is decidable. We can construct a DFA from the regular expression R using the subset construction algorithm. Then, we can check if the input string w is accepted by the constructed DFA. Therefore, this problem is decidable.
II. Given a context-free grammar G, is L(G) = ∅?
This problem is undecidable. This is because the emptiness problem for context-free grammars is undecidable. There is no algorithm that can determine whether a given context-free grammar generates any strings or not.
III. Given a context-free grammar G, is L(G) = Σ* for some alphabet Σ?
This problem is undecidable. This is because the universal language problem for context-free grammars is undecidable. There is no algorithm that can determine whether a given context-free grammar generates all possible strings over some alphabet.
IV. Given a Turing machine M and a string w, is w ∈ L(M)?
This problem is undecidable. This is because the halting problem for Turing machines is undecidable. There is no algorithm that can determine whether a given Turing machine halts on a given input or not.
Conclusion:
From the analysis above, we can conclude that the undecidable decision problems are:
- 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)?
Therefore, the correct option is III and IV only (option D).
Explanation:
In the given question, we are asked to determine which of the decision problems are undecidable. To analyze this, let's look at each option one by one.
I. Given a regular expression R and a string w, is w ∈ L(R)?
This problem is decidable. We can construct a DFA from the regular expression R using the subset construction algorithm. Then, we can check if the input string w is accepted by the constructed DFA. Therefore, this problem is decidable.
II. Given a context-free grammar G, is L(G) = ∅?
This problem is undecidable. This is because the emptiness problem for context-free grammars is undecidable. There is no algorithm that can determine whether a given context-free grammar generates any strings or not.
III. Given a context-free grammar G, is L(G) = Σ* for some alphabet Σ?
This problem is undecidable. This is because the universal language problem for context-free grammars is undecidable. There is no algorithm that can determine whether a given context-free grammar generates all possible strings over some alphabet.
IV. Given a Turing machine M and a string w, is w ∈ L(M)?
This problem is undecidable. This is because the halting problem for Turing machines is undecidable. There is no algorithm that can determine whether a given Turing machine halts on a given input or not.
Conclusion:
From the analysis above, we can conclude that the undecidable decision problems are:
- 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)?
Therefore, the correct option is III and IV only (option D).
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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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer?
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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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer?.
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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer?.
Solutions for 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 onlyb)II and III onlyc)II, III and IV onlyd)III and IV onlyCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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