bounded maximization

bounded maximization

The proof involved in showing that

functions

obtained via bounded minimizing (

http://planetmath.org/BoundedMinimization

)

primitive recursive functions

are themselves primitive recursive can be used to show the primitive recursiveness of another family of functions derived from existing primitive recursive functions via a

similar

technique, called

bounded maximization

.

Definition

. Let

f

:

ℕ

m

+

1

→

ℕ

be a function. For each

y

∈

ℕ

, set

A

f

⁢

(

𝒙

,

y

)

:=

{

z

∈

ℕ

∣

z

≤

y

⁢

and

⁢

f

⁢

(

𝒙

,

z

)

=

0

}

.

Define

f

b

⁢

m

⁢

a

⁢

x

⁢

(

𝒙

,

y

)

:=

{

max

⁡

A

f

⁢

(

𝒙

,

y

)

if

⁢

A

f

⁢

(

𝒙

,

y

)

≠

∅

,

0

otherwise.

f

b

⁢

m

⁢

a

⁢

x

is called the function obtained from

f

by

bounded maximization

.

Proposition 1

.

If

f

:

N

m

+

1

→

N

is primitive recursive, so is

f

b

⁢

m

⁢

a

⁢

x

.

Proof.

The proof of this is very similar to the proof that

f

b

⁢

m

⁢

i

⁢

n

is primitive recursive in the parent entry. The initial set up is the same: define

g

:=

sgn

∘

f

, where

sgn

is the sign function. So

g

is primitive recursive.

Next, define

h

:

ℕ

m

+

2

→

ℕ

by

h

⁢

(

𝒙

,

y

,

z

)

=

g

⁢

(

𝒙

,

y

⁢

-

˙

⁢

z

)

. So

h

, and therefore its

bounded product

:

h

p

⁢

(

𝒙

,

y

,

z

)

:=

∏

i

=

0

z

h

⁢

(

𝒙

,

y

,

i

)

are primitive recursive.

h

p

has the following

property

: given

y

,

•

if

k

is the largest number less than or equal to

y

such that

f

⁢

(

𝒙

,

k

)

=

0

, then

h

p

⁢

(

𝒙

,

y

,

z

)

:=

{

1

if

⁢

z

<

y

-

k

,

0

otherwise.

•

if no such

k

exists, then

h

p

⁢

(

𝒙

,

y

,

z

)

=

1

, for all

(

𝒙

,

y

,

z

)

∈

ℕ

m

+

2

.

As a result, the bounded sum

(

h

p

)

s

⁢

(

𝒙

,

y

,

z

)

:=

∑

i

=

0

z

h

p

⁢

(

𝒙

,

y

,

i

)

,

and in particular, the function

g

*

⁢

(

𝒙

,

y

)

:=

(

h

p

)

s

⁢

(

𝒙

,

y

,

y

)

, are primitive recursive. After some simplification,

g

*

becomes

g

*

⁢

(

𝒙

,

y

)

:=

{

y

-

k

if

⁢

k

=

max

⁡

A

f

⁢

(

𝒙

,

y

)

⁢

exists

,

s

⁢

(

y

)

otherwise.

Finally, the function

g

*

*

⁢

(

𝒙

,

y

)

:=

y

⁢

-

˙

⁢

g

*

⁢

(

𝒙

,

y

)

, which is just

f

b

⁢

m

⁢

a

⁢

x

⁢

(

𝒙

,

y

)

, is primitive recursive too.
∎

Example

. Using bounded maximization, we can show that

q

⁢

(

x

,

y

)

, the

quotient

of

x

÷

y

, is primitive recursive. When

y

=

0

, we set

q

⁢

(

x

,

y

)

=

0

First note that

q

⁢

(

x

,

y

)

is the largest integer

z

less than or equal to

x

such that

z

⁢

y

≤

x

. Let

A

⁢

(

y

,

x

)

=

{

z

∈

ℕ

∣

z

⁢

y

≤

x

}

. Then

A

⁢

(

y

,

x

)

can be rewritten as

{

z

∈

ℕ

∣

z

≤

x

⁢

and

⁢

sgn

⁡

(

y

⁢

z

⁢

-

˙

⁢

x

)

=

0

}

.

Define

f

:

ℕ

3

→

ℕ

by

f

⁢

(

x

,

y

,

t

)

=

sgn

⁡

(

y

⁢

t

⁢

-

˙

⁢

x

)

. Then

A

f

⁢

(

x

,

y

,

t

)

=

{

z

∈

ℕ

∣

z

≤

t

⁢

and

⁢

sgn

⁡

(

y

⁢

z

⁢

-

˙

⁢

x

)

=

0

}

.

Since

f

is primitive recursive, so is

f

b

⁢

m

⁢

a

⁢

x

⁢

(

x

,

y

,

t

)

. Since

A

⁢

(

x

,

y

)

=

A

f

⁢

(

x

,

y

,

x

)

, the quotient

q

⁢

(

x

,

y

)

may be defined as

f

b

⁢

m

⁢

a

⁢

x

⁢

(

x

,

y

,

x

)

, and therefore is primitive recursive.

With

q

⁢

(

x

,

y

)

, we may define

rem

⁡

(

x

,

y

)

, the

remainder

of

x

÷

y

, as

x

⁢

-

˙

⁢

y

⁢

q

⁢

(

x

,

y

)

, which is easily seen to be primitive recursive.

Remark

. Please see this entry (

http://planetmath.org/ExamplesOfPrimitiveRecursiveFunctions

) for an alternative way of showing that

q

⁢

(

x

,

y

)

and

rem

⁡

(

x

,

y

)

are primitive recursive without using bounded maximization. In the alternative method, one shows that

rem

⁡

(

x

,

y

)

is primitive recursive first. In

addition

,

rem

⁡

(

x

,

0

)

:=

0

in the alternative method, as opposed to

x

discussed here.

Title

bounded maximization

Canonical

name

BoundedMaximization

Date of creation

2013-03-22 19:05:37

Last modified on

2013-03-22 19:05:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D20

Related topic

BoundedMinimization

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