linear erasing

linear erasing

It is well-known that, among all of the

language

families in the

Chomsky hierarchy

, the family

𝒮

of

context-sensitive languages

is the only one that is not

closed under

arbitrary

homomorphisms

. Nevertheless,

𝒮

is shown to be closed under a more restricted class of homomorphisms, namely the

λ

-free homomorphisms. Question: can we enlarge this class of homomorphisms so that

𝒮

is still closed under the larger class? The answer is yes.

Definition

. Let

L

be a language over an

alphabet

Σ

,

h

a homomorphism over

Σ

, and

k

a non-negative integer.

h

is said to be a

k

-

linear erasing

on

L

if for any word

u

∈

L

, we have

|

u

|

≤

k

⁢

|

h

⁢

(

u

)

|

,

where

|

u

|

stands for the length of

u

.

It is clear that if

h

is a

k

-linear erasing on

L

, then it is a

m

-linear erasing for any

m

≥

k

. Also, if

h

is a

0

-linear erasing on

L

, then

L

is either

{

λ

}

, or the

empty set

∅

. In

addition

, if

h

is a

k

-linear erasing on

L

, and

L

′

⊆

L

, then it is a

k

-linear erasing on

L

′

. Consequently, any

λ

-free homomorphism is a

k

-linear erasing on any

L

over

Σ

, for any

k

≥

1

.

However, the notion of linear erasing is language dependent. For example, let

Σ

=

{

a

,

b

,

c

}

. Let

L

1

=

{

a

n

⁢

b

n

∣

n

≥

0

}

and

L

2

=

{

a

n

⁢

c

n

∣

n

≥

0

}

. Suppose

h

is the homomorphism on

Σ

*

with

h

⁢

(

a

)

=

λ

,

h

⁢

(

b

)

=

b

2

and

h

⁢

(

c

)

=

c

. Then

h

is a

1

-linear erasing on

L

1

, and a

2

-linear erasing on

L

2

.

Definition

Let

ℒ

be a family of languages over

Σ

. Then

ℒ

is said to be

closed under linear erasing

if for any

L

∈

ℒ

, and any homomorphism

h

which is a

k

-linear erasing on

L

for some

k

≥

0

, then

h

⁢

(

L

)

∈

ℒ

.

Clearly, if

ℒ

is closed under homomorphism, it is closed under linear erasing, and thus the families of

regular

(

http://planetmath.org/RegularLanguage

),

context-free

, and type-0 languages are all closed under linear erasing. We also have the following:

Theorem 1

.

The family

S

of context-sensitive languages is closed under linear erasing.

Remark

. The

theorem

above can be generalized. Call a

substitution

s

over

Σ

a

k

-linear erasing on a language

L

if

|

u

|

≤

k

⁢

|

v

|

for any

v

∈

s

⁢

(

u

)

. If

L

is context-sensitive such that

s

⁢

(

u

)

is context-sensitive for each

u

∈

L

, then

s

⁢

(

L

)

is context-sensitive provided that

s

is a

k

-linear erasing on

L

.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

2

J.E. Hopcroft, J.D. Ullman,

Formal Languages and Their

Relation

to

Automata

, Addison-Wesley, (1969).

Title

linear erasing

Canonical name

LinearErasing

Date of creation

2013-03-22 18:58:54

Last modified on

2013-03-22 18:58:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Synonym

limited erasing

Related topic

RestrictedHomomorphism

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