L = {w#x | w,x ∈{0,1}∗ and Turing Machine Mw with input x visits every non-end state at least once}

I believe this problem is undecidable.

My proof would consist of me reducing L to a Halting Problem; however, the structure of these proofs seem incosistent and inconclusive. I would appreciate if I could have some constructive feedback.

My understanding of the proof:

  1. Create R which decides L
  2. Construct algorithm S, which decides Atm (which we reduce to L)
  3. Contradiction because Atm is undecidable

This is the skeleton to most Turing Machine dicidibility related proofs I have read on the internet.

  • $\begingroup$ I suggest reviewing other similar proofs that were given in class. $\endgroup$ – Yuval Filmus Jun 26 '17 at 15:32
  • $\begingroup$ unfortunately as i said the structure of these proofs seem incosistent and inconclusive $\endgroup$ – Peter Bangert Jun 26 '17 at 15:53
  • 1
    $\begingroup$ You need to reduce the halting problem to $L$: you're saying that, if I could decide $L$, I could decide the halting problem for Turing machines. You seem to know what you need to do, so go ahead and do it: figure out a way of deciding the halting problem using a supposed ability to decide $L$. $\endgroup$ – David Richerby Jun 26 '17 at 17:44
  • $\begingroup$ "reducing L to a Halting Problem" does not allow you to state anything about the decidability of $L$. You need to reduce in the opposite direction. $\endgroup$ – chi Jun 26 '17 at 18:04
  • $\begingroup$ @chi, reduction to Halting problem leads to undecibility of initial problem, which is needed to be shown. $\endgroup$ – rus9384 Jun 26 '17 at 20:47

It's undecidable because you could use it to decide halting.

Because the number of states in a Turing machine is finite you can enumerate all $2^n$ combinations of possible non-end states. Run the algorithm against all of them.

If the algorithm returns yes for a group of states that includes an original end state then the machine halts.

Halting problem is undecidable so the algorithm cannot be executed by a Turing machine.


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