Is the language of TMs that halt on some string recognizable?

Question

I would like to show that the following language is recognizable: $$L:= \{ \langle M \rangle \mid M \text{ is a TM that halts on some string}\}.$$

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:

$$HALT_\epsilon:= \{ \langle M \rangle \mid M \text{ is a TM that halts on } \epsilon \},$$

where $\epsilon$ is the empty string. We can reduce $HALT$ on $x$ to $HALT_\epsilon$ by a reduction $F(\langle M, x\rangle) = \langle M' \rangle $, 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

$$HALT:= \{ \langle M,x \rangle \mid M \text{ is a TM that halts on } x \}.$$

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 $\langle M \rangle:$

1. Construct $N = $ "On input $x:$
   1. Run $M$ in parallel on all inputs $y\in \Sigma^*$.
   2. If $M$ halts on any $y$ then accept, otherwise loop."
2. Query the oracle to determine whether $\langle N \rangle \in HALT_\epsilon$.
3. 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 $\langle M \rangle \notin L$?

Answer 1

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 $\mathbb{N}$):

1. Simulate one step of $M$ on $1$.
2. Simulate two steps of $M$ on $1$ and $2$ each.
3. Simulate three steps on $1$, $2$ and $3$ each.

$\quad\vdots$

• 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.

Answer 2

Use nondeterminism. Given $M$, guess a string $w \in \mathcal{L}(M)$, simulate $M$ on $w$ (use your trusty universal Turing machine as a subroutine) and accept if $M$ accepts $w$.