linguistic:paraphrasing, stylistic:style_transfer
Recognizing partial solutions

There are many programs that, for some inputs, return a correct answer to the halting problem, while for other inputs they do not return an answer at all. However the problem "given program p, is it a partial halting solver" (in the sense described) is at least as hard as the halting problem.To see this, assume that there is an algorithm PHSR ("partial halting solver recognizer") to do that. Then it can be used to solve the halting problem,as follows:To test whether input program x halts on y, construct a program p that on input (x, y) reports true and diverges on all other inputs.Then test p with PHSR.

The above argument is a reduction of the halting problem to PHS recognition, and in the same manner, harder problems such as halting on all inputs can also be reduced, implying that PHS recognition is not only undecidable, but higher in the arithmetical hierarchy, specifically Π 2 0 {\displaystyle \Pi _{2}^{0}} -complete.