Partial Function

Write for partial functions from to . This means that not all inputs in have an output in . We write for the set of values which does have an output in .
We also write:

and use terminology that converges or diverges on input .

Given a Register Machine and , we can define a Partial Function by:

1. iff does not halt on
2. iff halts on with register content
   Note that is necessarily Computable.