Is there a $\Sigma^0_3$ variant of the halting problem?

In terms of the arithmetical hierarchy, the halting problem is known to be $\Sigma^0_1$-complete, and the so-called universal halting problem, is the problem of determining whether a given computer program will halt for every input, is known to be $\Pi^0_2$-complete.

Is there a natural $\Sigma^0_3$-complete variant of the halting problem?

Here are some natural problems which are complete for $\Sigma^0_3$:

• Given $\langle M \rangle$, is $L(M)$ cofinite?

• Given $\langle M \rangle$, is $L(M)$ recursive?

• Given $\langle M \rangle$, is $L(M)$ regular?

• Given $\langle M \rangle$, is $L(M)$ context-free?

In all of these, $L(M)$ is the set of inputs on which $M$ halts.

See these lecture notes.

StackExchange Computer Science Q#141215, answer score: 7