TM AS A COMPUTER OF INTEGER FUNCTIONS

TM halts (whether or not in a final state) with output on the tape.

What if is UNDEFINED?

Total Recursive Functions

f is defined for all inputs.
has a TM M that computes f and halts on all inputs for f.

e.g.

Partial Recursive Functions

f may be undefined for some inputs.
can be computed by a TM.

e.g.