Reduce ATM to REGULAR_TM

Question

Consider $\mathsf{REGULAR_{TM}} = \{\langle M \rangle \mid \text{$M$ is a TM and $L(M)$ is a regular language}\}$.

Let $S$ be the following algorithm, which solves $\mathsf{A_{TM}}$: “On input $\langle M, w \rangle$, where $M$ is a TM and $w$ is a string:

1. Construct the following TM $M_2$:

   $M_2$ = “On input $x$:

   1. If $x$ has the form $0^n 1^n$, accept.
   2. If $x$ does not have this form, run $M$ on input $w$ and accept if $M$ accepts $w$.”

2. Run $R$ (a machine solving $\mathsf{REGULAR_{TM}}$) on input $\langle M_2 \rangle$.

3. If $R$ accepts, accept; if $R$ rejects, reject.”

I am trying to understand this proof, but I am really confused with step 1 of $M_2$.

The book says "$M_2$ works by automatically accepting all strings in $\{0^n1^n \mid n ≥ 0\}$. In addition, if $M$ accepts $w$, $M_2$ accepts all other strings."

But I don't understand how could $M_2$ accept all strings. If $x$ is not of the form $0^n 1^n$ and $M$ does not accept $$, wouldn't $L( M_2 )$ be empty?. Or can we assume there are many different $x$ as an input to $M_2$?

Answer 1

First, note that $M$ and $w$ in $M_2$ are fixed. So whatever is the input, $M$ always accepts or rejects $w$. Second, note that $M_2$ is not built to run.

Then:

You want to map $<M,w>$ to a regular language $L$ if and only if $M$ accepts $w$. Fix $L$, in that case you choose $\Sigma^*$. However, if $M$ rejects $w$ you don't want $L$ regular (otherwise $R$ will accepts), so you choose $L'\subset L$ such that $L' = \{0^n1^n\}$ which is not regular.

Now you have two cases:

1) if $M$ rejects $w$, then $M_2$ will accepts only the strings of the form $0^n1^n$, then $R$ will reject.

2) if $M$ accepts $w$, then $M_2$ will accepts all the strings of the form $0^n1^n$ and all other strings in $\Sigma^*-0^n1^n$, so $R$ will accept.

However in poor words, it consists to map an "yes istance" of $<M,w>$ to a single "yes instance" of $L$. Let me do an example:

You want to prove that $L=\{M |\ if\ M\ accepts\ w\ then\ M\ accepts\ flip(w) \}$, where $flip(w)$ is the bitwise of all the string $w$ (For example if $w$ = $100$ then $flip(w) = 011$.), is undecidable with a reduction from $A_{tm}$.

Suppose $L$ is decidable, then exists a $TM$ $R$ that decides $L$. So we can build a new $TM$ $S$ that decides $A_{tm}$.

We will doing something like this:

$S$ on input $<M,w>$: "build a new $TM$ where $M$ and $w$ are constants. Call it $N_{<M,w>}$, then run $R$ on input $N_{<M,w>}$ and do what $R$ does."

Inside $N_{<M,w>}$ we have our reduction. Take in mind we want this:

$M$ accepts $w$ if and only if $R$ accepts $L(N_{<M,w>})$

First, we fix $L(N_{<M,w>})$ = $\{100,011\}$ (This is just an "yes instance" of $L(N_{<M,w>})$

Now our $N_{<M,w>}$

$N_{<M,w>}$ on input $x$:

if $x$ = $100$ then accepts

else if $x$=011 then run $M$ on $w$ and do what $M$ does.

Then "run" $R$ on $N_{<M,w>}$ and $S$ will accept if $R$ does:

if $M$ rejects $w$, the only string could be accepted by $N_{<M,w>}$ is $100$ then both $R$ and $S$ will reject.

Otherwise if $M$ accepts $w$, $N_{<M,w>}$ will accept both $100$ and $011$ then both $R$ and $S$ will accept.

So $S$ can decide $A_{TM}$ which is impossible, the "error" was to suppose the existance of $R$.