bounded minimization

bounded minimization

One useful way of generating more

primitive recursive functions

from existing ones is through what is known as

bounded summation

and

bounded product

. Given a primitive recursive function

f

:

ℕ

m

+

1

→

ℕ

, define two

functions

f

s

,

f

p

:

ℕ

m

+

1

→

ℕ

as follows: for

𝒙

∈

ℕ

m

and

y

∈

ℕ

:

f

s

⁢

(

𝒙

,

y

)

:=

∑

i

=

0

y

f

⁢

(

𝒙

,

i

)

f

p

⁢

(

𝒙

,

y

)

:=

∏

i

=

0

y

f

⁢

(

𝒙

,

i

)

These are easily seen to be primitive recursive, because they are defined by

primitive recursion

. For example,

f

s

⁢

(

𝒙

,

0

)

=

f

⁢

(

𝒙

,

0

)

,

and

f

s

⁢

(

𝒙

,

n

+

1

)

=

g

⁢

(

𝒙

,

n

,

f

s

⁢

(

𝒙

,

n

)

)

,

where

g

⁢

(

𝒙

,

n

,

y

)

=

add

⁡

(

f

⁢

(

𝒙

,

n

)

,

y

)

, which is primitive recursive by

functional

composition

.

Definition

. We call

f

s

and

f

p

functions obtained from

f

by

bounded sum

and

bounded product

respectively.

Using bounded summation and bounded product, another useful class of primitive recursive functions can be generated:

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

⁢

i

⁢

n

⁢

(

𝒙

,

y

)

:=

{

min

⁡

A

f

⁢

(

𝒙

,

y

)

if

⁢

A

f

⁢

(

𝒙

,

y

)

≠

∅

,

s

⁢

(

y

)

otherwise.

f

b

⁢

m

⁢

i

⁢

n

is called the function obtained from

f

by

bounded minimization

, and is usually denoted

μ

z

≤

y

(

f

(

𝒙

,

z

)

=

0

)

.

Proposition 1

.

If

f

:

N

m

+

1

→

N

is primitive recursive, so is

f

b

⁢

m

⁢

i

⁢

n

.

Proof.

Define

g

:=

sgn

∘

f

. Then

g

⁢

(

𝒙

,

y

)

:=

{

0

if

⁢

f

⁢

(

𝒙

,

y

)

=

0

,

1

otherwise.

As

f

is primitive recursive, so is

g

, since the sign function

sgn

is primitive recursive (see this entry (

http://planetmath.org/ExamplesOfPrimitiveRecursiveFunctions

)).

Next, the function

g

p

obtained from

g

by bounded product has the following

properties

:

•

if

g

p

⁢

(

𝒙

,

y

)

=

1

, then

g

p

⁢

(

𝒙

,

z

)

=

1

for all

z

<

y

,

•

if

g

p

⁢

(

𝒙

,

y

)

=

0

, then

g

p

⁢

(

𝒙

,

z

)

=

0

for all

z

≥

y

.

Finally, the function

(

g

p

)

s

obtained from

g

p

by bounded sum has the property that, when applied to

(

𝒙

,

y

)

, counts the number of

z

≤

y

such that

g

p

⁢

(

𝒙

,

z

)

=

1

. Based on the property of

g

p

, this count is then exactly the least

z

≤

y

such that

g

p

⁢

(

𝒙

,

z

)

=

1

. This means that

(

g

p

)

s

=

f

b

⁢

m

⁢

i

⁢

n

for all

(

𝒙

,

y

)

∈

ℕ

m

+

1

. Since

g

p

is primitive recursive, so is

(

g

p

)

s

, or that

f

b

⁢

m

⁢

i

⁢

n

is primitive recursive.
∎

In fact, if

f

is a (total)

recursive function

, so is

f

b

⁢

m

⁢

i

⁢

n

, because all of the derived functions in the proof above

preserve

primitive recursiveness as well as totalness.

Remarks

.

•

In the definition of bounded minimization, if we take the

y

out, then we arrive at the notion of

unbounded minimization

, or just

minimization

. The

proposition

above shows that the set

𝒫

⁢

ℛ

of primitive recursive functions is

closed under

bounded minimization. However,

𝒫

⁢

ℛ

is not closed under minimization. The

closure

of

𝒫

⁢

ℛ

under minimization is the set

ℛ

of recursive functions (total or not).

•

It is not hard to show that

ℰ

⁢

ℛ

, the set of all

elementary recursive functions

, is closed under bounded minimization.

Title

bounded minimization

Canonical

name

BoundedMinimization

Date of creation

2013-03-22 19:05:18

Last modified on

2013-03-22 19:05:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D20

Synonym

bounded sum

Related topic

UnboundedMinimization

Related topic

RecursiveFunction

Related topic

BoundedMaximization

Defines

bounded summation

Defines

bounded product

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