Even if there is no general algorithm to decide if any program will halt, but there could be properties or meta-questions about the programs that is decidable. For example, given program $A$ and a program $B$ that is obtainable from program $A$ by adding in finite steps of calculations that halts. So if $A$ halts, $B$ definitely halts (and vice versa). A trivial example is $B = A$.
The question "Do $A$ and $B$ have the same behaviour?" is decidable (they either both halt or run forever), even if we do not know whether $A$ or $B$ halts or not.
My question: Is it true that, for any program $A$, there is always a non-trivial program $B$ such that there is a decidable meta-question about the duo $(A, B)$? (Non-trivial means $B ≠ A$ and $B$ is not obtainable from $A$ via adding extra finite steps that halts. In general, it means $B\neq f(A)$, where $f$ is a computable function.)
I wonder if there is a field of research on this type of meta-problems?