This is a problem that I think is a reduction from the halting problem. I have on two separate practice exams statements that,
Given a Turing machine T and a string w, to determine whether T will halt when started on input w.
Given a Turing machine T and a string w, to determine whether T will halt when started on a blank tape.
The answer key states that these problems are both partially decidable, which is astonishing to me given that they are reductions from the halting problem, which is undecidable.
If there was a Turing maching M, that would halt if it's input was another Turing machine T that also halted on an arbitrary string, would that not violate the halting problem? Why are the two above statements considered to be partially decidable?