elementary recursive function

elementary recursive function

Informally,

elementary recursive functions

are

functions

that can be obtained from functions encountered in elementary schools:

addition

,

multiplication

,

subtraction

, and

division

, using basic

operations

such as substitutions and finite

summation

and

product

. Before stating the formal

definition

, the following notations are used:

ℱ

=

⋃

{

F

k

∣

k

∈

ℕ

}

, where for each

k

∈

ℕ

⁢

,

⁢

F

k

=

{

f

∣

f

:

ℕ

k

→

ℕ

}

.

Definition

. The set of

elementary recursive functions

, or

elementary functions

for short, is the smallest subset

ℰ

⁢

ℛ

of

ℱ

where:

1.

(addition)

add

∈

ℰ

⁢

ℛ

∩

F

2

, given by

add

⁡

(

m

,

n

)

:=

m

+

n

;

2.

(multiplication)

mult

∈

ℰ

⁢

ℛ

∩

F

2

, given by

mult

⁡

(

m

,

n

)

:=

m

⁢

n

;

3.

(

difference

)

diff

∈

ℰ

⁢

ℛ

∩

F

2

, given by

diff

⁡

(

m

,

n

)

:=

|

m

-

n

|

;

4.

(quotient)

quo

∈

ℰ

⁢

ℛ

∩

F

2

, given by

quo

⁡

(

m

,

n

)

:=

[

m

/

n

]

, which is the largest non-negative

integer

z

such that

n

⁢

z

≤

m

(by convention,

quo

⁡

(

0

,

0

)

:=

1

);

5.

(

projection

)

p

m

k

∈

ℰ

⁢

ℛ

∩

F

k

, where

m

≤

k

, given by

p

m

k

⁢

(

n

1

,

…

,

n

k

)

:=

n

m

;

6.

ℰ

⁢

ℛ

is

closed under

composition: If

{

g

1

,

…

,

g

m

}

⊆

ℰ

⁢

ℛ

∩

F

k

and

h

∈

ℰ

⁢

ℛ

∩

F

m

, then

f

∈

ℰ

⁢

ℛ

∩

F

k

, where

f

⁢

(

n

1

,

…

,

n

k

)

=

h

⁢

(

g

1

⁢

(

n

1

,

…

,

n

k

)

,

…

,

g

m

⁢

(

n

1

,

…

,

n

k

)

)

;

7.

ℰ

⁢

ℛ

is closed under

bounded sum

: if

f

∈

ℰ

⁢

ℛ

∩

F

k

, then

f

s

∈

ℰ

⁢

ℛ

∩

F

k

, where

f

s

⁢

(

𝒙

,

y

)

:=

∑

i

=

0

y

f

⁢

(

𝒙

,

i

)

;

8.

ℰ

⁢

ℛ

is closed under bounded product: if

f

∈

ℰ

⁢

ℛ

∩

F

k

, then

f

p

∈

ℰ

⁢

ℛ

∩

F

k

, where

f

p

⁢

(

𝒙

,

y

)

:=

∏

i

=

0

y

f

⁢

(

𝒙

,

i

)

.

Examples

.

•

The initial functions in the definition of

primitive recursive functions

are elementary:

(a)

The

zero function

z

⁢

(

x

)

is

diff

⁡

(

x

,

x

)

.

(b)

The

constant function

const

1

⁡

(

x

)

:=

1

is

quo

⁡

(

x

,

x

)

.

(c)

The successor function

s

⁢

(

x

)

can be obtained by substituting (by composition) the constant function

const

1

and the projection function

p

1

1

, into the addition function

add

⁡

(

p

1

1

⁢

(

x

)

,

const

1

⁡

(

x

)

)

.

•

Multiplication

mult

in 2 above may be removed from the definition, since

mult

⁡

(

x

,

y

)

=

diff

⁡

(

f

⁢

(

x

,

y

)

,

p

1

2

⁢

(

x

,

y

)

)

,

where

⁢

f

⁢

(

x

,

y

)

:=

∑

i

=

0

y

p

1

2

⁢

(

x

,

i

)

•

Many other basic functions, such as the restricted subtraction,

exponential function

, the

i

-th prime function, are all elementary. One may replace the difference function in 3 above by the restricted subtraction without changing

ℰ

⁢

ℛ

.

Remarks

•

Consider the set

𝒫

⁢

ℛ

of primitive recursive functions. The functions in the first five groups above are all in

𝒫

⁢

ℛ

. In addition,

𝒫

⁢

ℛ

is closed under the operations in 6, 7, and 8 above, we see that

ℰ

⁢

ℛ

⊆

𝒫

⁢

ℛ

, since

ℰ

⁢

ℛ

, as defined, is the smallest such set.

•

Furthermore,

ℰ

⁢

ℛ

≠

𝒫

⁢

ℛ

. For example, the super-exponential function, given by

f

⁢

(

x

,

0

)

=

m

, and

f

⁢

(

x

,

n

+

1

)

=

exp

⁡

(

m

,

f

⁢

(

x

,

n

)

)

, where

m

>

1

, can be shown to be non-elementary.

•

In addition, it can be shown that

ℰ

⁢

ℛ

is the set of primitive recursive functions that can be obtained from the zero function, the successor function, and the projection functions via composition, and no more than three applications of

primitive recursion

.

•

By taking the

closure

of

ℰ

⁢

ℛ

with respect to unbounded minimization, one obtains

ℛ

, the set of all

recursive functions

(partial or total). In fact, by replacing bounded sum and bounded product with unbounded minimization, and the difference function with restricted subtraction, one obtains

ℛ

.

Title

elementary recursive function

Canonical

name

ElementaryRecursiveFunction

Date of creation

2013-03-22 19:06:48

Last modified on

2013-03-22 19:06:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D20

Synonym

elementary

Related topic

BoundedRecursion

Defines

elementary recursive

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