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Which of the following statements are undecidable?
For a given Turing Machine M,
For a given Turing Machine M,
- a)does M halt on an empty input tape
- b)does M halt for anly inputs at all?
- c)is L(M) regular? Context free? Turing decidable?
- d)all of the mentioned
Correct answer is option 'D'. Can you explain this answer?
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Which of the following statements are undecidable?For a given Turing M...
All of the following mentioned are immediate results of Rice’s theorem and thus, undecidable.
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Which of the following statements are undecidable?For a given Turing M...
Undecidable Statements in Turing Machines
To understand why option 'D' is the correct answer, let's analyze each statement individually and determine if it is undecidable or not.
a) Does M halt on an empty input tape?
This statement refers to the halting problem, which is known to be undecidable. The halting problem asks whether a given Turing machine halts on a specific input. The proof of this undecidability is based on a diagonalization argument by contradiction. Therefore, statement (a) is undecidable.
b) Does M halt for any inputs at all?
This statement is also a variation of the halting problem. It asks whether a given Turing machine halts on any input, regardless of its content. Since the halting problem is undecidable, this statement is also undecidable.
c) Is L(M) regular? Context free? Turing decidable?
This statement refers to the language recognized by Turing machine M. Let's analyze each part separately:
- Is L(M) regular?
Determining if a language is regular is decidable. There are algorithms, such as the pumping lemma or finite automata equivalence, that can be used to determine if a language is regular.
- Is L(M) context-free?
Determining if a language is context-free is also decidable. There are algorithms, such as the CYK algorithm or parsing techniques, that can be used to determine if a language is context-free.
- Is L(M) Turing decidable?
Determining if a language is Turing decidable is decidable. A language is Turing decidable if there exists a Turing machine that halts and accepts every string in the language, and halts and rejects every string not in the language. This can be determined by simulating the Turing machine on all possible inputs.
Therefore, statement (c) is not undecidable since it includes decidable questions about the language recognized by Turing machine M.
d) All of the mentioned
Since statements (a) and (b) are undecidable, and statement (c) is not undecidable, the correct answer is option 'D' - all of the mentioned statements are undecidable.
To understand why option 'D' is the correct answer, let's analyze each statement individually and determine if it is undecidable or not.
a) Does M halt on an empty input tape?
This statement refers to the halting problem, which is known to be undecidable. The halting problem asks whether a given Turing machine halts on a specific input. The proof of this undecidability is based on a diagonalization argument by contradiction. Therefore, statement (a) is undecidable.
b) Does M halt for any inputs at all?
This statement is also a variation of the halting problem. It asks whether a given Turing machine halts on any input, regardless of its content. Since the halting problem is undecidable, this statement is also undecidable.
c) Is L(M) regular? Context free? Turing decidable?
This statement refers to the language recognized by Turing machine M. Let's analyze each part separately:
- Is L(M) regular?
Determining if a language is regular is decidable. There are algorithms, such as the pumping lemma or finite automata equivalence, that can be used to determine if a language is regular.
- Is L(M) context-free?
Determining if a language is context-free is also decidable. There are algorithms, such as the CYK algorithm or parsing techniques, that can be used to determine if a language is context-free.
- Is L(M) Turing decidable?
Determining if a language is Turing decidable is decidable. A language is Turing decidable if there exists a Turing machine that halts and accepts every string in the language, and halts and rejects every string not in the language. This can be determined by simulating the Turing machine on all possible inputs.
Therefore, statement (c) is not undecidable since it includes decidable questions about the language recognized by Turing machine M.
d) All of the mentioned
Since statements (a) and (b) are undecidable, and statement (c) is not undecidable, the correct answer is option 'D' - all of the mentioned statements are undecidable.
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