Computability complexity and Languages- Fundamentals of Theoretical Computer Science

4. Computable Functions 31
Computability theory (also called recursion theory) studies the class of
partially computable functions. In order to justify the name, we need some
evidence that for every function which one can claim to be "computable"
on intuitive grounds, there really is a program of the language Y which
computes it. Such evidence will be developed as we go along.
We close this section with one final example of a program of .Y:
[A] X+-- X+ 1
IF X* 0 GOTO A
For this program 9', I/J.~ 1 >(x) is undefined for all x. So, the nowhere
defined function (see Chapter 1, Section 2) must be included in the class of
partially computable functions.
Exercises
1. Let 9' be the program
What is 1/J.~IJ(x )?
IF Xi= OGOTOA
[A] X+-- X+ 1
IF Xi= OGOTO A
[A] Y +-- Y + 1
2. The same as Exercise 1 for the program
[ B] IF X * 0 GOTO A
Z+-Z+1
IF Z * OGOTO B
[A] X+-X
3. The same as Exercise 1 for the empty program.
4. Let 9' be the program
[A]
Y+-XI
IFX 2 =0GOTO£
Y+-Y+1
Y+-Y+1
X 2 +-- X 2 - 1
GOTOA
What is I/J.!)>(r 1 )? l/l}}>(r 1 , r 2 )? I/Jj}>(r 1 , r 2 , r 3 )?
5. Show that for every partially computable function f(x 1 , ••• , xn), there
is a number m ~ 0 such that f is computed by infinitely many
programs of length m.