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Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.
(I) For an unrestricted grammar G and a string w, where w = L(G).
(II) Given a Turing Machine M, whether L(M) is regular.
(III) Given two grammars G1 and G2 whether L(G1) = L(G2).
(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.
Which one of the following statements is correct?
- a)(I) is undecidable
- b)(II) is undecidable
- c)(III) is undecidable
- d)(IV) is undecidable
Correct answer is option 'A,B,C'. Can you explain this answer?
Most Upvoted Answer
Consider the following problems L(G) denotes the language generated by...
Given,
Statement (I): Undecidable
For an unrestricted grammar G and a string w, where w = L(G).
Membership problem for RE → undecidable.
Statement (II): Undecidable
Given a turing Machine M, whether L(M) is regular.
Regularity problem for RE → undecidable.
Statement (III): Undecidable
Given two grammars G1 and G2 whether L(G1) = L(G2).
Equivalence problem for RE→ undecidable.
Statement (IV): Decidable
Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.
Since DPDA P exists for every NFA N and is equivalent to it, this problem is trivially decidable.
Hence, the correct options are (A), (B) and (C).
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Consider the following problems L(G) denotes the language generated by...
Understanding the Problems
The problems presented involve different types of grammars and machines in formal language theory. Let's analyze each one to understand why they are undecidable.
Problem I: Unrestricted Grammar Membership
- Statement: For an unrestricted grammar G and a string w, whether w is in L(G).
- Undecidability: This problem is undecidable because it can be reduced from the Halting Problem. There is no algorithm that can determine if a given string belongs to the language generated by an unrestricted grammar.
Problem II: Regular Language Acceptance by Turing Machines
- Statement: Given a Turing Machine M, whether L(M) is regular.
- Undecidability: This is also undecidable. There is no effective method to ascertain if the language accepted by a Turing Machine is regular, as it involves understanding the intricacies of the language generated beyond regular capabilities.
Problem III: Equivalence of Two Grammars
- Statement: Given two grammars G1 and G2, whether L(G1) = L(G2).
- Undecidability: This problem is undecidable for unrestricted grammars. The equivalence of context-free grammars is known to be undecidable, thus extending to unrestricted grammars.
Problem IV: NFA and PDA Language Acceptance
- Statement: Given an NFA N, whether there exists a deterministic PDA P such that N and P accept the same language.
- Decidability: This problem is decidable, as NFAs and PDAs can be transformed into equivalent forms with clear language acceptance criteria.
Conclusion
- As a result, options A, B, and C are correct as they pertain to undecidable problems, while option D is incorrect because it addresses a decidable scenario.
The problems presented involve different types of grammars and machines in formal language theory. Let's analyze each one to understand why they are undecidable.
Problem I: Unrestricted Grammar Membership
- Statement: For an unrestricted grammar G and a string w, whether w is in L(G).
- Undecidability: This problem is undecidable because it can be reduced from the Halting Problem. There is no algorithm that can determine if a given string belongs to the language generated by an unrestricted grammar.
Problem II: Regular Language Acceptance by Turing Machines
- Statement: Given a Turing Machine M, whether L(M) is regular.
- Undecidability: This is also undecidable. There is no effective method to ascertain if the language accepted by a Turing Machine is regular, as it involves understanding the intricacies of the language generated beyond regular capabilities.
Problem III: Equivalence of Two Grammars
- Statement: Given two grammars G1 and G2, whether L(G1) = L(G2).
- Undecidability: This problem is undecidable for unrestricted grammars. The equivalence of context-free grammars is known to be undecidable, thus extending to unrestricted grammars.
Problem IV: NFA and PDA Language Acceptance
- Statement: Given an NFA N, whether there exists a deterministic PDA P such that N and P accept the same language.
- Decidability: This problem is decidable, as NFAs and PDAs can be transformed into equivalent forms with clear language acceptance criteria.
Conclusion
- As a result, options A, B, and C are correct as they pertain to undecidable problems, while option D is incorrect because it addresses a decidable scenario.
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Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer?.
Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following problems L(G) denotes the language generated by a grammar G.L(M) denotes the language accepted by a machine M.(I) For an unrestricted grammar G and a string w, where w = L(G).(II) Given a Turing Machine M, whether L(M) is regular.(III) Given two grammars G1 and G2 whether L(G1) = L(G2).(IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language.Which one of the following statements is correct?a)(I) is undecidableb)(II) is undecidablec)(III) is undecidabled)(IV) is undecidableCorrect answer is option 'A,B,C'. Can you explain this answer?.
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