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I can't find how to prove the decibility with a reduction.

EDIT: I've tried the reduction from the halting problem and the aceptance problem. Stopping for at least one entry has infinite inputs (you have to check all possible inputs) but the halting problem only has one input for the TM.

I don't understand how can i formally define a machine that using a machine that checks all inputs solves all cases of the halting problem.

dipzza
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  • Try harder. Hint: the problem of deciding whether a Turing machine halts on empty input is undecidable. – Steven May 14 '20 at 11:19
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    @Steven Almost no one who is stuck on a problem is going to get help from "Try harder". – Caleb Stanford May 14 '20 at 12:40
  • What have you tried reducing from? What progress have you made? We're happy to help you understand the concepts but just solving exercise-style problems for you is unlikely to achieve that. You might find this page helpful in improving your question. – D.W. May 14 '20 at 17:44
  • @6005 Agreed, but they are going to benefit from the hint. The "try harder" part was mostly from being annoyed at the fact this sounds a lot like an exercise question that OP showed no effort or indication of having tried to solve by himself. – Steven May 14 '20 at 18:08
  • these notes are pretty helpful. – Jack Nov 18 '20 at 16:13

2 Answers2

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It is not decidable. You can prove that by reducing the Halting problem to it as follows: Given a tuple $(M,x)$ and the question whether $M$ halts on $x$, we can construct a Turing machine $M_x$ which simply simulates $M$ on $x$ irrespective of the actual input $y$ of $M_x$. It is clear that $M_x$ will halt on any input iff $M$ halts on $x$.

prime_hit
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I found the answer to my question in this similar problem Halting problem reducing to the blank tape halting problem You can just erase the 'for any input' problem using a Turing Machine that always erase the input and write w. This way you can make an equivalence between an instance (M, w) of the halting problem and an instance (M) of this problem.

dipzza
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