is a partial halting solver. Then there exists a partial halting solver recognizer, PHSR, guaranteed to terminate with an answer. Construct a program H: By construction, program H is also guaranteed to terminate with an answer. If PHSR recognizes the constructed program X as a partial halting solver, that means that P, the only input for which X produces a result, halts. If PHSR fails to recognize X, then it must be because P does not halt. Therefore H can decide whether an arbitrary program P halts; it solves the halting problem. Since this is impossible, then the program PHSR could not have existed as supposed. Therefore, it's not possible to decide whether any given program is a partial halting solver. Another example, H T , of a Turing machine which gives correct answers only for some instances of the halting problem can be described by the requirements that, if H T is started scanning a field which carries the first of a finite string of a consecutive "1"s, followed by one field with symbol "0" (i. e. a blank field), and followed in turn by a finite string of i consecutive "1"s, on an otherwise blank tape, then: H T halts for any such starting state, i. e. for any input of finite positive integers a and i ; H T halts on a completely blank tape if and only if the Turing machine represented by a does not halt when given the starting state and input represented by