computability – 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?