terminating reduction

terminating reduction

Let

X

be a set and

→

a

reduction

(

binary relation

) on

X

. A

chain

with respect to

→

is a

sequence

of

elements

x

1

,

x

2

,

x

3

,

…

in

X

such that

x

1

→

x

2

,

x

2

→

x

3

, etc… A chain with respect to

→

is usually written

x

1

→

x

2

→

x

3

→

⋯

→

x

n

→

⋯

.

The length of a chain is the

cardinality

of its underlying sequence. A chain is finite if its length is finite. Otherwise, it is

infinite

.

Definition

. A reduction

→

on a set

X

is said to be

terminating

if it has no

infinite chains

. In other words, every chain

terminates

.

Examples

.

•

If

→

is

reflexive

, or non-trivial

symmetric

, then it is never terminating.

•

Let

X

be the set of all positive integers greater than

1

. Define

→

on

X

so that

a

→

b

means that

a

=

b

⁢

c

for some

c

∈

X

. Then

→

is a terminating reduction. By the way,

→

is also a

normalizing reduction

.

•

In fact, it is easy to see that a terminating reduction is normalizing: if

a

has no normal form, then we may form an infinite chain starting from

a

.

•

On the other hand, not all normalizing reduction is terminating. A

canonical

example is the set of all non-negative integers with the reduction

→

defined by

a

→

b

if and only if

–

either

a

,

b

≠

0

,

a

≠

b

, and

a

<

b

,

–

or

a

≠

0

and

b

=

0

.

The infinite chain is given by

1

→

2

→

3

→

⋯

, so that

→

is not terminating. However,

n

→

0

for every positive integer

n

. Thus every integer has

0

as its normal form, so that

→

is normalizing.

Remarks

.

•

A reduction is said to be

convergent

if it is both terminating and

confluent

.

•

A

relation

is terminating iff the

transitive closure

of its

inverse

is

well-founded

.

To see this, first let

R

be terminating on the set

X

. And let

S

be the transitive closure of

R

-

1

. Suppose

A

is a non-empty subset of

X

that contains no

S

-

minimal elements

. Pick

x

0

∈

A

. Then we can find

x

1

∈

A

with

x

1

≠

x

0

, such that

x

1

⁢

S

⁢

x

0

. By the

assumption

on

A

, this process can be iterated indefinitely. So we have a sequence

x

0

,

x

1

,

x

2

,

…

such that

x

i

+

1

⁢

S

⁢

x

i

with

x

i

≠

x

i

+

1

. Since each pair

(

x

i

,

x

i

+

1

)

can be expanded into a finite chain with respect to

R

, we have produced an infinite chain containing elements

x

0

,

x

1

,

x

2

,

…

, contradicting the assumption that

R

is terminating.

On the other hand, suppose the transitive closure

S

of

R

-

1

is well-founded. If the chain

x

0

⁢

R

⁢

x

1

⁢

R

⁢

x

2

⁢

R

⁢

⋯

is infinite, then the set

{

x

0

,

x

1

,

x

2

,

…

}

has no

S

-minimal elements, as

x

i

⁢

S

⁢

x

j

whenever

i

>

j

, and

j

arbitrary.

•

The reflexive transitive closure of a terminating relation is a

partial order

.

A closely related

concept

is the

descending chain condition

(DCC). A reduction

→

on

X

is said to

satisfy

the

descending chain condition (DCC)

if the only infinite chains on

X

are those that are

eventually constant

. A chain

x

1

→

x

2

→

x

3

→

⋯

is eventually constant if there is a positive integer

N

such that for all

n

≥

N

,

x

n

=

x

N

. Every terminating relation satisfies DCC. The

converse

is obviously not true, as a

reflexive reduction

illustrates.

Another related concept is acyclicity. Let

→

be a reduction on

X

. A chain

x

0

→

x

1

→

⋯

⁢

x

n

is said to be cyclic if

x

i

=

x

j

for some

0

≤

i

<

j

≤

n

. This means that there is a “closed loop” in the chain. The reduction

→

is said to be

acyclic

if there are no cyclic chains with respect to

→

. Every terminating relation is acyclic, but not conversely. The usual strict inequality relation on the set of positive integers is an example of an acyclic but non-terminating relation.

Title

terminating reduction

Canonical name

TerminatingReduction

Date of creation

2013-03-22 17:57:41

Last modified on

2013-03-22 17:57:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Related topic

NormalizingReduction

Related topic

Confluence

Related topic

DiamondLemma

Defines

terminating

Defines

descending chain condition

Defines

DCC

Defines

convergent reduction

Defines

acyclic

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