I would like to show that the following language is recognizable: $$L:= \\{ \\la
ID: 652905 • Letter: I
Question
I would like to show that the following language is recognizable: $$L:= { langle M angle mid M ext{ is a TM that halts on some string}}.$$
How do I go about showing that this language is recognizable? I know that all recognizable languages are reducible to $HALT_epsilon$, so I figure if I can show that this language reduces to $HALT_epsilon$, then I am all set. I am defining $HALT_epsilon$ as follows:
$$HALT_epsilon:= { langle M angle mid M ext{ is a TM that halts on } epsilon },$$
where $epsilon$ is the empty string. We can reduce $HALT$ on $x$ to $HALT_epsilon$ by a reduction $F(langle M, x angle) = langle M' angle $, where $M'(y) = M(x)$. For this reduction, we just ignore the input string $y$, which we know will be $epsilon$ and just run $M$ on $x$ instead. Here, $HALT$ is defined as
$$HALT:= { langle M,x angle mid M ext{ is a TM that halts on } x }.$$
I tried leveraging a similar technique to show that $L$ is recognizable, but I could not come up with anything better than this (somewhat crazy) TM that has a $HALT_epsilon$ oracle:
$ D^{HALT_epsilon} =$ On input $langle M angle:$
Construct $N = $ "On input $x:$
Run $M$ in parallel on all inputs $yin Sigma^*$.
If $M$ halts on any $y$ then accept, otherwise loop."
Query the oracle to determine whether $langle N angle in HALT_epsilon$.
If the oracle answers YES, accept; if NO, reject.
Note: My notation for TM algorithms is based on "Theory of Computation" by Sipser. Step 2 for the definition of $N$ is a bit redundant, but in this type of context, is it okay to say something like "If $M$ halts on any $y$, then halt?"
I think all I have shown here is that $L$ is decidable relative to $HALT_epsilon$. I don't know if this implies that $L$ is recognizable. Can a Turing reduction be used in this manner to show that a language is recognizable? I'm confused as to what it means for a language to be recognizable. The task seems obvious if we go back to the definition: If some TM $R$ accepts strings in $L$, then $R$ recognizes $L$. So what if $R=D^{HALT_epsilon}$, and in the body of $D^{HALT_epsilon}$ we use some crazy reduction like $N$?
In general, to show recognizability, can we just come up with a reduction like $N$ that may or may not halt? Is it a problem that $N$ will never halt if $langle M angle otin L$?
Explanation / Answer
You can construct a recognizer following the same principle used for the recognizer for HALT. The only extra bit is how you check "all" inputs without getting stuck in a non-terminating computation.
An important technique you can use here is called dovetailing (expressed with inputs from $mathbb{N}$):
Simulate one step of $M$ on $1$.
Simulate two steps of $M$ on $1$ and $2$ each.
Simulate three steps on $1$, $2$ and $3$ each.
$quad dots$
Terminate and accept once any of the simulated computations terminates and accepts.
If there is a halting input of $M$, this dovetailed simulation certainly finds it after finite time. If there is none, it loops and is correct in doing so.
This is, in essence, your $N$ (with an explanation why it's actually a computable function). You don't need the rest of the reduction in order to show that $L$ is semi-decidable.