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I'm taking an introduction to computation theory class and we went over the chapter on undecidable problems and proving undecidability through reductions. I can't seem to grasp some of the simplest problems when it comes to reduction however.

For example there is the problem $\mbox{REGULAR}_{TM} = \{ \langle M \rangle \mid M\mbox{ is a Turing Machine and }L(M)\mbox{ is a regular language}\}$

Most of the time the answers involve creating your own auxiliary Turing Machine. For this example, my book decides to reduce from $A_{TM}$ my book uses $M_{aux}$. I show what $M_{aux}$ looks like within the decider $S$ for $A_{TM}$ they formulated ($R$ is the decider for $\mbox{REGULAR}_{TM}$):

$S = $ "On input $\langle M , w \rangle$, where $M$ is a TM and $w$ is a string:

  1. Construct the following TM $M_{aux}$:

    $M_{aux}$: "On input $x$:

    1. If $x$ has the form $0^n1^n$, then accept.
    2. If $x$ does not have this form, run $M$ on input $w$ and accept if $M$ accepts $w$."
  2. Run $R$ to on $\langle M_{aux} \rangle$.

  3. If $R$ accepts, accept. If $R$ rejects, reject."

I don't really understand where $x$ comes from and how $M_{aux}$ uses $x$. According to the book, $M_{aux}$ works by automatically accepting all strings in $\{0^n1^n\mid n \ge 0\}$. If $M$ accepts $w$, $M_{aux}$ accepts all other strings. But what if $M$ doesn't accept $w$ in the second step? In that case it looks like $M_{aux}$ would reject. What happens then?

My main problem is just that I don't understand how $M_{aux}$ runs and how the decider for $R$ uses it. I've looked over many other examples and I understand how to actually create proofs like this, but what I don't understand is the subroutine machine that they create, like $S$ does.

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The point is that the behavior of $M_{\rm aux}$ depends only on what $M$ does to $w$:

  • If $M$ accepts $w$, then the language $M_{\rm aux}$ accepts is regular (namely everything).
  • If $M$ rejects $w$ (such as if it doesn't terminate), then the language $M_{\rm aux}$ accepts is not regular (namely $\{0^n1^n\}$ which is easily shown not to be regular, for example by the Myhill-Nerode theorem).

The input $x$ doesn't come from anywhere, because $M_{\rm aux}$ is never actually run -- we're just feeding it to $R$ to be analyzed, in order that $R$ can tell us which of the two above cases we're in.

It's not particularly important that the two languages $M_{\rm aux}$ may recognize are precisely $0^n1^n$ and $\{0,1\}^*$ -- the important thing is that they ought to be classified differently by $R$, and therefore we can use $R$ (if it works as assumed) to learn things about $M$'s behavior of $w$ that we already know we can't know in general.

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