deletion operation on languages

deletion operation on languages

Let

Σ

be an

alphabet

and

u

,

v

be words over

Σ

. A

deletion

of

v

from

u

is a word of the form

u

1

⁢

u

2

, where

u

=

u

1

⁢

v

⁢

u

2

. If

w

is a deletion of

v

from

u

, then

u

is an

insertion

of

v

into

w

.

The deletion

of

v

from

u

is the set of all deletions of

v

from

u

, and is denoted by

u

⟶

v

.

For example, if

u

=

(

a

⁢

b

)

6

and

v

=

a

⁢

b

⁢

a

, then

u

⟶

v

=

{

(

b

⁢

a

)

4

⁢

b

,

a

⁢

b

⁢

(

b

⁢

a

)

3

⁢

b

,

(

a

⁢

b

)

2

⁢

(

b

⁢

a

)

2

⁢

b

,

(

a

⁢

b

)

3

⁢

b

⁢

a

⁢

b

,

(

a

⁢

b

)

4

⁢

b

}

.

More generally, given two

languages

L

1

,

L

2

over

Σ

, the

deletion

of

L

2

from

L

1

is the set

L

1

⟶

L

2

:=

⋃

{

u

⟶

v

∣

u

∈

L

1

,

v

∈

L

2

}

.

For example, if

L

1

=

{

(

a

⁢

b

)

n

∣

n

≥

0

}

and

L

2

=

{

a

n

⁢

b

n

∣

n

≥

0

}

, then

L

1

⟶

L

2

=

L

1

, and

L

2

⟶

L

1

=

L

2

. If

L

3

=

{

a

n

∣

n

≥

0

}

, then

L

2

⟶

L

3

=

a

m

b

n

∣

n

≥

m

≥

0

}

, and

L

3

⟶

L

2

=

L

3

.

A language

L

is said to be

deletion-closed

if

L

⟶

L

⊆

L

.

L

1

,

L

2

, and

L

3

from above are all deletion closed, as well as the most

obvious

example:

Σ

*

.

L

2

⟶

L

3

is not deletion closed, for

a

3

⁢

b

4

⟶

a

⁢

b

3

=

{

a

2

⁢

b

}

⊈

L

2

⟶

L

3

.

It is easy to see that arbitrary

intersections

of deletion-closed languages is deletion-closed.

Given a language

L

, the intersection of all deletion-closed languages containing

L

, denoted by

Del

⁡

(

L

)

, is called the

deletion-closure

of

L

. In other words,

Del

⁡

(

L

)

is the smallest deletion-closed language containing

L

.

The deletion-closure of a language

L

can be constructed from

L

, as follows:

L

0

:=

L

L

n

+

1

:=

L

n

⟶

(

L

n

∪

{

λ

}

)

L

′

:=

⋃

i

=

0

∞

L

i

Then

Del

⁡

(

L

)

=

L

′

.

For example, if

L

=

{

a

p

∣

p

⁢

is a prime number

}

, then

Del

⁡

(

L

)

=

{

a

n

∣

n

≥

0

}

, and if

L

=

{

a

m

⁢

b

n

∣

m

≥

n

≥

0

}

, then

Del

⁡

(

L

)

=

{

a

m

⁢

b

n

∣

m

,

n

≥

0

}

.

Remarks

.

•

If

L

1

,

L

2

are

regular languages

, so is

L

1

⟶

L

2

. In other words, the family

ℛ

of regular languages is

closed under

the deletion

binary operation

.

•

In

addition

,

ℛ

is closed under deletion closure: if

L

is

regular

, so is

Del

⁡

(

L

)

.

•

However,

ℱ

, the family of

context-free languages

, is neither closed under deletion, nor under deletion closure.

References

1

M. Ito,

Algebraic Theory of

Automata

and Languages

, World Scientific, Singapore (2004).

Title

deletion operation on languages

Canonical

name

DeletionOperationOnLanguages

Date of creation

2013-03-22 18:56:23

Last modified on

2013-03-22 18:56:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 68Q45

Synonym

deletion-closed

Synonym

deletion-closure

Related topic

ShuffleOfLanguages

Related topic

InsertionOperationOnLanguages

Defines

deletion

Defines

deletion closed

Defines

deletion closure

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