alternative characterization of primitive recursiveness

alternative characterization of primitive recursiveness

One useful feature regarding the extended notion of

primitive

recursiveness described in the parent entry is that it can be used to characterize the original notion of primitive recursiveness. As in the parent entry, we use the notation

𝒮

:=

{

f

:

ℕ

m

→

ℕ

∣

any

m

≥

1

}

,

𝒱

:=

{

f

:

ℕ

m

→

ℕ

n

∣

any

m

,

n

≥

1

}

.

In

addition

, let

𝒱

⁢

(

m

,

n

)

:=

{

f

∈

𝒱

∣

f

:

ℕ

m

→

ℕ

n

}

.

Finally, denote

𝒫

⁢

ℛ

the set of

primitive recursive functions

in the traditional sense (as a subset of

𝒮

), and

𝒫

⁢

ℛ

′

the set of

primitive recursive vector-valued functions

(a subset of

𝒱

). It is clear that

𝒫

⁢

ℛ

=

𝒫

⁢

ℛ

′

∩

𝒮

.

Let

𝒟

be the smallest subset of

𝒱

such that

1.

𝒟

contains the zero function

z

, the successor function

s

, and the projection functions

p

m

k

(see the definition of primitive recursive functions for more detail),

2.

𝒟

is

closed under

functional

composition

in

𝒱

,

3.

𝒟

is closed under

extension

of coordinates: that is, if

f

1

,

…

,

f

n

∈

𝒱

⁢

(

m

,

1

)

are in

𝒟

, so is

f

:=

(

f

1

,

…

,

f

n

)

∈

𝒱

⁢

(

m

,

n

)

, given by

f

⁢

(

𝒙

)

=

(

f

1

⁢

(

𝒙

)

,

…

,

f

n

⁢

(

𝒙

)

)

,

4.

𝒟

is closed under iterated composition: if

f

∈

𝒱

⁢

(

m

,

m

)

is in

𝒟

, and so is

g

∈

𝒱

⁢

(

m

+

1

,

n

)

, given by

g

⁢

(

𝒙

,

n

)

=

f

n

⁢

(

𝒙

)

(where

f

0

is the

identity function

).

We now state the

characterization

.

Proposition 1

.

𝒫

⁢

ℛ

′

=

𝒟

.

Proof.

First, observe that condition 1 is satisfied by both

𝒫

⁢

ℛ

′

and

𝒟

. To see that

𝒟

⊆

𝒫

⁢

ℛ

′

, note that condition 3 is just the definition in the parent entry, and conditions 2 and 4 are discussed and proved, also in the parent entry. So we have one

inclusion

. To see the other inclusion

𝒫

⁢

ℛ

′

⊆

𝒟

, we need to verify the two closure properties:

1.

functional composition (in

𝒮

): suppose

g

1

,

…

,

g

m

∈

𝒱

⁢

(

k

,

1

)

and

h

∈

𝒱

⁢

(

m

,

1

)

are in

𝒟

, we want to show that

f

:=

h

⁢

(

g

1

,

…

,

g

m

)

∈

𝒱

⁢

(

k

,

1

)

is in

𝒟

too. First, form

g

=

(

g

1

,

…

,

g

m

)

. Then

g

∈

𝒟

by extension of coordinates. Then

f

=

h

∘

g

∈

𝒟

by functional composition (in

𝒱

).

2.

primitive recursion

: suppose

g

∈

𝒱

⁢

(

k

,

1

)

and

h

∈

𝒱

⁢

(

k

+

2

,

1

)

are both in

𝒟

, we want to show that

f

∈

𝒱

⁢

(

k

+

1

,

1

)

given by

f

⁢

(

𝒙

,

0

)

=

g

⁢

(

𝒙

)

and

f

⁢

(

𝒙

,

n

+

1

)

=

h

⁢

(

𝒙

,

n

,

f

⁢

(

𝒙

,

n

)

)

is in

𝒟

too. First, define a function

H

∈

𝒱

⁢

(

k

+

2

,

k

+

2

)

by

H

⁢

(

𝒙

,

y

,

z

)

:=

(

𝒙

,

s

⁢

(

y

)

,

h

⁢

(

𝒙

,

y

,

z

)

)

.

Then

H

is formed by extension of coordinates via the projection functions

p

i

k

+

2

(with

i

=

1

,

…

,

k

) producing the first

k

coordinates (the coordinates of

𝒙

), the function

s

∘

p

k

+

1

k

+

2

producing the

k

+

1

st coordinate

s

⁢

(

y

)

, and

h

producing the last coordinate. Since each of the coordinate functions is in

D

, so is

H

.

Next, the function

F

∈

𝒱

⁢

(

k

+

3

,

k

+

2

)

given by

F

⁢

(

𝒙

,

y

,

z

,

m

)

=

H

m

⁢

(

𝒙

,

y

,

z

)

is in

D

by iterated composition. We now verify by

induction

on

n

that

F

⁢

(

𝒙

,

0

,

g

⁢

(

𝒙

)

,

n

)

=

(

𝒙

,

n

,

f

⁢

(

𝒙

,

n

)

)

.

–

F

(

𝒙

,

0

,

g

(

𝒙

)

,

0

)

=

(

𝒙

,

0

,

g

(

𝒙

)

, and its third coordinate is

f

⁢

(

𝒙

,

0

)

, as desired.

–

Suppose now that

F

⁢

(

𝒙

,

0

,

g

⁢

(

𝒙

)

,

n

)

=

(

𝒙

,

n

,

f

⁢

(

𝒙

,

n

)

)

. Then

F

⁢

(

𝒙

,

0

,

g

⁢

(

𝒙

)

,

n

+

1

)

=

H

⁢

(

𝒙

,

n

,

f

⁢

(

𝒙

,

n

)

)

=

(

𝒙

,

s

⁢

(

n

)

,

h

⁢

(

𝒙

,

n

,

f

⁢

(

𝒙

,

n

)

)

)

=

(

𝒙

,

n

+

1

,

f

⁢

(

𝒙

,

n

+

1

)

)

.

As a result,

f

⁢

(

𝒙

,

n

)

=

p

k

+

2

k

+

2

∘

F

⁢

(

𝒙

,

0

,

g

⁢

(

𝒙

)

,

n

)

is in

𝒟

also.

Therefore,

𝒫

⁢

ℛ

′

⊆

𝒟

, and the proof is

complete

.
∎

Remark

. According to the characterization above, one sees that primitive recursion is in a sense a special form of iterated composition. The above characterization is helpful in proving, among other things, that every

URM-computable

function is

recursive

, and that the

Ackermann function is not primitive recursive

.

Title

alternative characterization of primitive recursiveness

Canonical name

AlternativeCharacterizationOfPrimitiveRecursiveness

Date of creation

2013-03-22 19:06:44

Last modified on

2013-03-22 19:06:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Result

Classification

msc 03D20

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