alternative characterizations of recursive functions

alternative characterizations of recursive functions

The class of

recursive functions

may be characterized by considerably weaker conditions than those given in the entry “recursive function (

http://planetmath.org/RecursiveFunction

)” of this encyclopaedia. This entry will discuss several such

characterizations

.

Criteria 2 and 3 in the list may be replaced by the considerably weaker criterion:

2’) The successor function

S

:

ℤ

+

→

ℤ

+

defined as

S

⁢

(

x

)

=

x

+

1

is a recursive function.

By means of a

pairing function

, the

definition

may be simplified considerably. Using such a

function

and its

inverses

, the set of recursive functions of

m

variables

may be put in

one-to-one correspondence

with recursive functions of

n

variables for any pair of non-zero positive integers

m

and

n

. Hence one can focus attention on recursive functions of a small fixed number of variables. One characterization of recursive functions of not more than two variables is the following:

The class of recursive functions is the smallest class of positive integer valued functions of not more than two positive integers which

satisfies

the following criteria:

1’

The

constant function

c

:

ℤ

+

→

ℤ

+

defined by

c

⁢

(

x

)

=

1

for all

x

∈

ℤ

+

is a recursive function.

2’

The successor function

S

:

ℤ

+

→

ℤ

+

defined as

S

⁢

(

x

)

=

x

+

1

is a recursive function.

3’

The projection functions

I

1

1

:

ℤ

+

→

ℤ

+

,

I

1

2

:

ℤ

+

2

→

ℤ

+

, and

I

2

2

:

ℤ

+

2

→

ℤ

+

defined as

I

1

1

⁢

(

x

)

=

x

I

1

2

⁢

(

x

,

y

)

=

x

I

2

2

⁢

(

x

,

y

)

=

y

are recursive functions.

4’

If

a

:

ℤ

+

2

→

ℤ

+

,

b

:

ℤ

+

2

→

ℤ

+

,

c

:

ℤ

+

2

→

ℤ

+

,

d

:

ℤ

+

→

ℤ

+

, and

e

:

ℤ

+

→

ℤ

+

are recursive functions, then

f

:

ℤ

+

2

→

ℤ

+

,

g

:

ℤ

+

→

ℤ

+

, and

h

:

ℤ

+

2

→

ℤ

+

, defined by

f

⁢

(

x

,

y

)

=

d

⁢

(

a

⁢

(

x

,

y

)

)

g

⁢

(

x

)

=

a

⁢

(

d

⁢

(

x

)

,

e

⁢

(

y

)

)

h

⁢

(

x

,

y

)

=

a

⁢

(

b

⁢

(

x

,

y

)

,

c

⁢

(

x

,

y

)

)

are recursive functions.

5’

If

f

:

ℤ

+

→

ℤ

+

,

g

:

ℤ

+

→

ℤ

+

are recursive functions, then the function

h

:

ℤ

+

2

→

ℤ

+

defined by the

recursion

h

⁢

(

n

+

1

,

x

)

=

g

⁢

(

h

⁢

(

n

,

x

)

)

with the initial condition

h

⁢

(

0

,

x

)

=

f

⁢

(

x

)

is a recursive function.

6’

If

f

:

ℤ

+

2

→

ℤ

+

is a recursive function then

g

:

ℤ

+

→

ℤ

+

is a recursive function, where

g

⁢

(

x

)

is defined to equal

y

if there exists a

y

∈

ℤ

+

such that

(a)

f

⁢

(

0

,

x

)

,

f

⁢

(

1

,

x

)

,

…

⁢

f

⁢

(

y

,

x

)

are all defined,

(b)

f

⁢

(

z

,

x

)

≠

0

when

1

≤

z

<

y

, and

(c)

f

⁢

(

y

,

x

)

=

0

.

Otherwise,

g

⁢

(

x

)

is undefined.

The criterion 5’ may be shown to follow from the remaining criteria, and hence it may be dropped.

By further exploiting the marvelous

properties

of the pairing function, criterion 6’ may be replaced by the following:

6”) If

f

:

ℤ

+

→

ℤ

+

is a recursive function then

g

:

ℤ

+

→

ℤ

+

is a recursive function, where

g

⁢

(

x

)

is defined to equal

y

if there exists a

y

∈

ℤ

+

such that

a

f

⁢

(

0

)

,

f

⁢

(

1

)

,

…

⁢

f

⁢

(

y

)

are all defined,

b

f

⁢

(

z

)

≠

x

when

1

≤

z

<

y

, and

c

f

⁢

(

y

)

=

x

.

Otherwise,

g

⁢

(

x

)

is undefined.

The

operation

introduced in this new criterion is called

minimized inversion

and will be denoted as

g

=

f

-

1

. Note that there is no conflict with the usual notion of inverse of a function because, if

f

is invertible, the minimized inverse of

f

is the same as the inverse of

f

in the usual sense; otherwise the notion of minimized inverse extends the definition of inverse to a larger class of functions.

Finally, by taking these ideas even further, Czirmaz showed that recursive functions of a single variable had the following simple characterization:

The class of recursive functions is the smallest class of positive integer valued functions of a positive integer which satisfies the following criteria:

1”

The constant function

c

:

ℤ

+

→

ℤ

+

defined by

c

⁢

(

x

)

=

1

for all

x

∈

ℤ

+

is a recursive function.

2”

The successor function

S

:

ℤ

+

→

ℤ

+

defined as

S

⁢

(

x

)

=

x

+

1

is a recursive function.

3”

The function

Q

:

ℤ

+

→

ℤ

+

defined as

Q

⁢

(

x

)

=

x

-

⌈

x

⌉

2

is a recursive function. In words,

Q

⁢

(

x

)

is the

difference

between

x

and the largest square number smaller than

x

.

4”

If

f

:

ℤ

+

→

ℤ

+

and

g

:

ℤ

+

→

ℤ

+

are recursive functions, then

h

:

ℤ

+

→

ℤ

+

is a recursive function, where

h

⁢

(

x

)

=

f

⁢

(

g

⁢

(

x

)

)

6”

If

f

:

ℤ

+

→

ℤ

+

is a recursive function then

g

:

ℤ

+

→

ℤ

+

is a recursive function, where

g

⁢

(

x

)

is defined to equal

y

if there exists a

y

∈

ℤ

+

such that

a

f

⁢

(

0

)

,

f

⁢

(

1

)

,

…

⁢

f

⁢

(

y

)

are all defined,

b

f

⁢

(

z

)

≠

x

when

1

≤

z

<

y

, and

c

f

⁢

(

y

)

=

x

.

Otherwise,

g

⁢

(

x

)

is undefined.

Title

alternative characterizations of recursive functions

Canonical

name

AlternativeCharacterizationsOfRecursiveFunctions

Date of creation

2013-03-22 14:34:42

Last modified on

2013-03-22 14:34:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 03D20

Defines

minimized inversion

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