μ-operator

μ

-operator

μ

on predicates

Let a

property

on non-negative integers be given. Informally, the

μ

-

operator

looks for the smallest number satisfying the given property. The

μ

-operator is also known as the (

unbounded

)

minimization operator

or the (unbounded) search operator. Formally,

Definition

. Let

Φ

⁢

(

𝒙

,

y

)

be an

(

m

+

1

)

-ary

predicate

(property) over

ℕ

, the set of natural numbers (

0

included here), with

m

a non-negative integer (

𝒙

is

m

-ary). Define

A

Φ

⁢

(

𝒙

)

:=

{

y

∈

ℕ

∣

Φ

⁢

(

𝒙

,

y

)

}

,

The

μ

-operator

on

Φ

is given by

μ

⁢

y

⁢

Φ

⁢

(

𝒙

,

y

)

:=

{

min

⁡

A

Φ

⁢

(

𝒙

)

if

⁢

A

Φ

⁢

(

𝒙

)

≠

∅

,

undefined

otherwise.

The notation

μ

⁢

y

⁢

Φ

⁢

(

𝒙

,

y

)

reads “the smallest

y

such that

Φ

⁢

(

𝒙

,

y

)

is satisfied”. Note that

y

is used both as a

variable

and a number that the variable

y

represents.

Note that

μ

⁢

y

⁢

Φ

⁢

(

𝒙

,

y

)

is a

partial function

on

𝒙

(

y

is a

bounded

variable). In other words, the

μ

-operator is a

function

that takes an

m

+

1

-ary predicate to an

m

-ary partial function. When

m

=

0

,

μ

is either an integer, or

∅

.

For example, suppose

Φ

⁢

(

x

,

y

)

is the property

(

x

-

y

)

⁢

(

x

+

y

)

≥

10

. Then

μ

⁢

y

⁢

Φ

⁢

(

x

,

y

)

=

0

iff

x

≥

4

, and undefined otherwise.

The reason why the

μ

-operator is called the search operator comes from

recursive function

theory. The search for the smallest

y

such that

Φ

⁢

(

𝒙

,

y

)

begins with testing for

Φ

⁢

(

𝒙

,

0

)

. If the test fails (

Φ

⁢

(

𝒙

,

0

)

is false), then testing for

Φ

⁢

(

𝒙

,

1

)

,

Φ

⁢

(

𝒙

,

2

)

,

Φ

⁢

(

𝒙

,

3

)

,

⋯

are successively performed. The testing stops when a

y

with

Φ

⁢

(

𝒙

,

y

)

is found. This

y

is also the smallest

y

satisfying

Φ

. Nevertheless, the testing can conceivably go on indefinitely, hence the name

unbounded

. There is also a bounded version of

μ

-

operation

:

Definition

. Let

Φ

⁢

(

𝒙

,

y

)

be given as above. Define

A

Φ

⁢

(

𝒙

,

y

)

:=

{

z

∈

ℕ

∣

Φ

⁢

(

𝒙

,

z

)

⁢

and

⁢

z

≤

y

}

.

The

bounded

μ

-operator

on

Φ

is given by

μ

⁢

z

≤

y

⁢

Φ

⁢

(

𝒙

,

z

)

:=

{

min

⁡

A

Φ

⁢

(

𝒙

,

y

)

if

⁢

A

Φ

⁢

(

𝒙

,

y

)

≠

∅

,

y

+

1

otherwise.

Thus the bounded

μ

-operator takes an

(

m

+

1

)

-ary predicate (on

(

𝒙

,

z

)

, where

z

is a

free variable

) to an

(

m

+

1

)

-ary

total function

(on

(

𝒙

,

y

)

, as

z

is now bounded by

μ

).

μ

on total functions

The

μ

operator can also be defined on functions. We first discuss the case of

μ

on total functions.

Definition

. Given a total

(

m

+

1

)

-ary function

f

:

ℕ

m

+

1

→

ℕ

, define

A

f

⁢

(

𝒙

)

:=

{

y

∈

ℕ

∣

f

⁢

(

𝒙

,

y

)

=

0

}

,

The

μ

-operator

on

f

is given by

μ

⁢

y

⁢

f

⁢

(

𝒙

,

y

)

:=

{

min

⁡

A

f

⁢

(

𝒙

)

if

⁢

A

f

⁢

(

𝒙

)

≠

∅

,

undefined

otherwise.

This definition is actually

equivalent

to the one regarding predicates, in the following sense: given a total function

f

with arity

m

+

1

, define predicate

Φ

f

⁢

(

𝒙

,

y

)

of arity

m

+

1

, as “

f

⁢

(

𝒙

,

y

)

=

0

”. Then

μ

⁢

y

⁢

f

⁢

(

𝒙

,

y

)

=

μ

⁢

y

⁢

Φ

f

⁢

(

𝒙

,

y

)

.

Conversely

, given an

(

m

+

1

)

-ary predicate

Φ

⁢

(

𝒙

,

y

)

, define an

(

m

+

1

)

-ary function

f

Φ

⁢

(

𝒙

,

y

)

:=

1

⁢

-

˙

⁢

χ

Φ

⁢

(

𝒙

,

y

)

, where

χ

Φ

is the

characteristic function

of

Φ

. Then

μ

⁢

y

⁢

Φ

⁢

(

𝒙

,

y

)

=

μ

⁢

y

⁢

f

Φ

⁢

(

𝒙

,

y

)

.

Note also that

μ

⁢

f

may be partial even if

f

is total, since it is possible

f

⁢

(

𝒙

,

y

)

≠

0

for all

y

, and the search will go on indefinitely. For example, let

f

⁢

(

x

,

z

,

y

)

=

x

2

+

z

2

-

y

2

if

x

2

+

z

2

≥

y

2

, and

1

otherwise. Clearly,

f

is a total function. It is easy to see that

μ

⁢

y

⁢

f

⁢

(

3

,

4

,

y

)

=

5

, while

μ

⁢

y

⁢

f

⁢

(

1

,

2

,

y

)

is undefined.

μ

on partial functions

The definition of the

μ

-operator on total functions can be generalized to partial functions. However, in recursive functions theory, an additional condition is imposed in order to make the

generalization

.

Definition

. Given a partial

(

m

+

1

)

-ary function

f

:

ℕ

m

+

1

→

ℕ

, define

A

f

⁢

(

𝒙

)

:=

{

y

∈

ℕ

∣

f

⁢

(

𝒙

,

y

)

=

0

⁢

and

⁢

(

𝒙

,

z

)

∈

dom

⁡

(

f

)

⁢

for all

⁢

z

≤

y

}

,

The

μ

-operator

on

f

is given by

μ

⁢

y

⁢

f

⁢

(

𝒙

,

y

)

:=

{

min

⁡

A

f

⁢

(

𝒙

)

if

⁢

A

f

⁢

(

𝒙

)

≠

∅

,

undefined

otherwise.

The extra condition that

f

⁢

(

𝒙

,

z

)

be defined for all

z

≤

y

is needed in order to avoid situations where testing for

f

⁢

(

𝒙

,

z

)

=

0

gets stuck in an infinite loop (in a

Turing machine

or a URM) because

f

⁢

(

𝒙

,

z

)

is undefined, before

y

is ever reached for testing. If we drop this extra condition, then it is possible to find a

partial recursive function

f

such that

μ

⁢

f

is not

recursive

.

Remarks

.

•

Bounded

μ

-operator may also be defined on functions. In the case of partial functions, the definition can be given as follows: let

f

⁢

(

𝒙

,

y

)

be an

(

m

+

1

)

-ary partial function, with

A

f

⁢

(

𝒙

,

y

)

:=

{

z

∈

ℕ

∣

f

⁢

(

𝒙

,

z

)

=

0

,

z

≤

y

,

and

⁢

(

𝒙

,

t

)

∈

dom

⁡

(

f

)

⁢

for all

⁢

t

≤

z

}

,

then

μ

⁢

z

≤

y

⁢

f

⁢

(

𝒙

,

z

)

:=

{

min

⁡

A

f

⁢

(

𝒙

,

y

)

if

⁢

A

f

⁢

(

𝒙

,

y

)

≠

∅

,

y

+

1

if

⁢

A

f

⁢

(

𝒙

,

y

)

=

∅

⁢

and

⁢

(

𝒙

,

t

)

∈

dom

⁡

(

f

)

⁢

for all

⁢

t

≤

y

undefined

otherwise.

•

Given the set

𝒫

⁢

ℛ

of

primitive recursive functions

, one obtains the set

ℛ

of recursive functions by taking the

closure

of

𝒫

⁢

ℛ

with respect to the application of the

μ

-operator. Furthermore, if

f

∈

ℛ

, so is

μ

⁢

f

∈

ℛ

, and it can be shown that any recursive function can be obtained from primitive recursive functions by no more than one application of the

μ

-operator. This is known as the Kleene

normal form

theorem

.

•

With respect to the bounded

μ

-operator, any primitive recursive function (or predicate) stays primitive recursive after an application of the bounded

μ

, and any

total recursive function

stays total after an application of the bounded

μ

.

Title

μ

-operator

Canonical

name

muoperator

Date of creation

2013-03-22 19:05:47

Last modified on

2013-03-22 19:05:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D20

Synonym

minimization operator

Synonym

minimization-operator

Synonym

search operator

Synonym

search-operator

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