GATE | GATE-CS-2001 | Question 32

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? 

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


Answer: (B)

Explanation:
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.

 
Thus, option (B) is the answer.

 
Please comment below if you find anything wrong in the above post.


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