Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the following problem X.Given a Turi...
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Consider the following problem X.
Given a Turing machine M over the input alphabet Σ, any state q of M And a word w∈Σ*, does the computation of M on w visit the state q?
Q. Which of the following statements about X is correct?
- a)X is decidable
- b)X is undecidable but partially decidable
- c)X is undecidable and not even partially decidable
- d)X is not a decision problem
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Consider the following problem X.Given a Turing machine M over the inp...
This problem is a State Entry Problem. State entry problem can be reduced to halting problem.
We construct a turing machine M with final state ‘q’. We run a turing machine R (for state entry problem) with inputs : M, q, w .
We give ‘w’ as input to M.
If M halts in the final state ‘q’ then R accepts the input. So, the given problem is partially decidable. If M goes in an infinite loop then M can not output anything. So, R rejects the input. So, the given problem becomes undecidable.
We give ‘w’ as input to M.
If M halts in the final state ‘q’ then R accepts the input. So, the given problem is partially decidable. If M goes in an infinite loop then M can not output anything. So, R rejects the input. So, the given problem becomes undecidable.
Thus, option (B) is the answer.
Please comment below if you find anything wrong in the above post.
Most Upvoted Answer
Consider the following problem X.Given a Turing machine M over the inp...
Understanding the Problem X
The problem X revolves around determining whether a given Turing machine M, when provided with an input word w, visits a specific state q during its computation. This inquiry involves the behavior of Turing machines, which are fundamental to the theory of computation.
Decidability of X
- Undecidable Nature: The problem X is undecidable because it can be framed in terms of the Halting Problem. The Halting Problem states that there is no algorithm that can determine whether any arbitrary Turing machine halts on a given input. Since determining if M visits state q can be related to whether M halts, X inherits this undecidability.
Partial Decidability
- Partially Decidable Status: While X is undecidable, it is partially decidable. This means that there exists a Turing machine that can accept inputs for which M visits state q, but it may not halt for all inputs where state q is not visited. Thus, we can recognize some cases but not all.
Conclusion
- Correct Answer: The answer option B is correct: "X is undecidable but partially decidable." This reflects the complexities involved in the behavior of Turing machines and highlights the limitations of computational decision-making.
Summary
- The problem X is a significant concept in theoretical computer science, illustrating the boundaries of what can be computed or decided about Turing machines. Understanding this helps in grasping the limitations inherent in algorithmic processes.
The problem X revolves around determining whether a given Turing machine M, when provided with an input word w, visits a specific state q during its computation. This inquiry involves the behavior of Turing machines, which are fundamental to the theory of computation.
Decidability of X
- Undecidable Nature: The problem X is undecidable because it can be framed in terms of the Halting Problem. The Halting Problem states that there is no algorithm that can determine whether any arbitrary Turing machine halts on a given input. Since determining if M visits state q can be related to whether M halts, X inherits this undecidability.
Partial Decidability
- Partially Decidable Status: While X is undecidable, it is partially decidable. This means that there exists a Turing machine that can accept inputs for which M visits state q, but it may not halt for all inputs where state q is not visited. Thus, we can recognize some cases but not all.
Conclusion
- Correct Answer: The answer option B is correct: "X is undecidable but partially decidable." This reflects the complexities involved in the behavior of Turing machines and highlights the limitations of computational decision-making.
Summary
- The problem X is a significant concept in theoretical computer science, illustrating the boundaries of what can be computed or decided about Turing machines. Understanding this helps in grasping the limitations inherent in algorithmic processes.
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