insertion operation on languages

insertion operation on languages

Let

Σ

be an

alphabet

, and

u

,

v

words over

Σ

. An

insertion

of

u

into

v

is a word of the form

v

1

⁢

u

⁢

v

2

, where

v

=

v

1

⁢

v

2

. The

concatenation

of two words is just a special case of insertion. Also, if

w

is an insertion of

u

into

v

, then

v

is a

deletion

of

u

from

w

.

The insertion

of

u

into

v

is the set of all insertions of

v

into

u

, and is denoted by

v

⊳

u

.

The notion of insertion can be extended to

languages

. Let

L

1

,

L

2

be two languages over

Σ

. The insertion of

L

1

into

L

2

, denoted by

L

1

⊳

L

2

, is the set of all insertions of words in

L

1

into words in

L

2

. In other words,

L

1

⊳

L

2

=

⋃

{

u

⊳

v

∣

u

∈

L

1

,

v

∈

L

2

}

.

So

u

⊳

v

=

{

u

}

⊳

{

v

}

.

A language

L

is said to be

insertion closed

if

L

⊳

L

⊆

L

. Clearly

Σ

*

is insertion closed, and arbitrary

intersection

of insertion closed languages is insertion closed. Given a language

L

, the

insertion closure

of

L

, denoted by

Ins

⁡

(

L

)

, is the smallest insertion closed language containing

L

. It is clear that

Ins

⁡

(

L

)

exists and is unique.

More to come…

The

concept

of insertion can be generalized. Instead of the insertion of

u

into

v

taking place in one location in

v

, the insertion can take place in several locations, provided that

u

must also be broken up into pieces so that each

individual

piece goes into each inserting location. More precisely, given a positive integer

k

, a

k

-insertion of

u

into

v

is a word of the form

v

1

⁢

u

1

⁢

⋯

⁢

v

k

⁢

u

k

⁢

v

k

+

1

where

u

=

u

1

⁢

⋯

⁢

u

k

and

v

=

v

1

⁢

⋯

⁢

v

k

+

1

. So an insertion is just a

1

-insertion.

The

k

-insertion

of

u

into

v

is the set of all

k

-insertions of

u

into

v

, and is denoted by

u

⊳

[

k

]

v

. Clearly

⊳

[

1

]

⁣

=

⁣

⊳

.

Example

. Let

Σ

=

{

a

,

b

}

, and

u

=

a

⁢

b

⁢

a

,

v

=

b

⁢

a

⁢

b

, and

w

=

b

⁢

a

⁢

b

⁢

a

⁢

b

⁢

a

. Then

•

w

is an insertion of

u

into

v

, as well as an insertion of

v

into

u

, for

w

=

v

⁢

u

⁢

λ

=

λ

⁢

v

⁢

u

.

•

w

is also a

2

-insertion of

u

into

v

:

–

the decompositions

u

=

(

a

⁢

b

)

⁢

(

a

)

and

v

=

(

b

)

⁢

(

a

⁢

b

)

⁢

λ

–

or the decompositions

u

=

λ

⁢

u

and

v

=

λ

⁢

v

⁢

λ

produce

(

b

)

⁢

(

a

⁢

b

)

⁢

(

a

⁢

b

)

⁢

(

a

)

⁢

λ

=

λ

⁢

λ

⁢

v

⁢

u

⁢

λ

=

w

.

•

Finally,

w

is also a

2

-insertion of

v

into

u

, as the decompositions

u

=

λ

⁢

u

⁢

λ

and

v

=

v

⁢

λ

produce

λ

⁢

v

⁢

u

⁢

λ

⁢

λ

=

w

.

•

u

⊳

v

=

{

a

⁢

b

⁢

a

⁢

b

⁢

a

⁢

b

,

b

⁢

a

⁢

b

⁢

a

⁢

a

⁢

b

,

b

⁢

a

⁢

a

⁢

b

⁢

a

⁢

b

,

b

⁢

a

⁢

b

⁢

a

⁢

b

⁢

a

}

.

From this example, we observe that a

k

-insertion is a

(

k

+

1

)

-insertion, and every

k

-insertion of

u

into

v

is a

(

k

+

1

)

-insertion of

v

into

u

. As a result,

u

⊳

[

k

]

v

⊆

(

u

⊳

[

k

+

1

]

v

)

∩

(

v

⊳

[

k

+

1

]

u

)

.

As before, given languages

L

1

and

L

2

,

the

k

-insertion

of

L

1

into

L

2

, denoted by

L

1

⊳

[

k

]

L

2

, is the union of all

u

⊳

[

k

]

v

, where

u

∈

L

1

and

v

∈

L

2

.

Remark

. Some

closure properties

regarding insertions: let

ℛ

be the family of

regular languages

, and

ℱ

the family of

context-free languages

. Then

ℛ

is

closed under

⊳

[

k

]

, for each positive integer

k

.

ℱ

is closed

⊳

[

k

]

only when

k

=

1

. If

L

1

∈

ℛ

and

L

2

∈

ℱ

, then

L

1

⊳

[

k

]

L

2

and

L

2

⊳

[

k

]

L

1

are both in

ℱ

. The proofs of these closure properties can be found in the reference.

References

1

M. Ito,

Algebraic Theory of

Automata

and Languages

, World Scientific, Singapore (2004).

Title

insertion operation on languages

Canonical name

InsertionOperationOnLanguages

Date of creation

2013-03-22 18:56:53

Last modified on

2013-03-22 18:56:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 68Q45

Synonym

insertion-closed

Synonym

insertion-closure

Related topic

DeletionOperationOnLanguages

Related topic

ShuffleOfLanguages

Defines

insertion

Defines

insertion closed

Defines

insertion closure

Defines

k

-insertion

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