Chapter 3
Recursive Function Theory
In this chapter, we will take a brief look at the recursive function theory. We will only consider
functions that map positive integers to positive integers, and in particular, only functions which
produce a single output i.e.,
f
:
N
n
→
N
. Let us first define some operations that we can do on
functions:
Definition 4 Composition:
Given the following functions,
f
1
(
x
1
, x
2
,
· · ·
, x
n
)
,
f
2
(
x
1
, x
2
,
· · ·
, x
n
)
,
.
.
.
f
k
(
x
1
, x
2
,
· · ·
, x
n
)
, and
g
(
x
1
, x
2
,
· · ·
, x
k
)
we define the function
h
(
x
1
, x
2
,
· · ·
, x
n
)
to be the composition of
f
1
, f
2
,
· · ·
, f
k
with
g
as follows:
h
((
x
1
, x
2
,
· · ·
, x
n
) =
g
(
f
1
(
x
1
, x
2
,
· · ·
, x
n
)
,
· · ·
, f
k
(
x
1
, x
2
,
· · ·
, x
n
))
.
A function which has output values defined for each input value is called a
Total
function. An
example of such a function is
f
(
x
) = 2
x
.
A function is not Total, if for some input values, no
output value is defined. The following function is not a Total function:
f
(
x
) =
x
2
if x even
undefined
if x odd
Composition preserves Totality, i.e. if