Kuroda normal form

Kuroda normal form

Just like every

context-free grammar

can be “normalized” or “standardized” in one of the so-called

normal forms

(

http://planetmath.org/ChomskyNormalForm

), so can a

context-sensitive grammar

, and in fact arbitrary

grammar

, be normalized. The particular normalization discussed in this entry is what is known as the

Kuroda normal form

.

A formal grammar

G

=

(

Σ

,

N

,

P

,

σ

)

is in

Kuroda normal form

if its productions have one of the following forms:

1.

A

→

a

,

2.

A

→

B

⁢

C

,

3.

A

⁢

B

→

C

⁢

D

.

where

a

∈

Σ

-

N

and

A

,

B

,

C

,

D

∈

N

.

Note: Sometimes the form

A

→

B

, where

B

∈

N

is added to the list above. However, we may remove a production of this form by replacing all

occurrences

of

B

by

A

in every production of

G

.

The usefulness of the Kuroda normal forms is captured in the following result:

Theorem 1

.

A grammar is length-increasing iff it is

equivalent

to a grammar in Kuroda normal form.

Note that the third production form

A

⁢

B

→

C

⁢

D

may be replaced by the following productions

1.

A

⁢

B

→

A

⁢

X

2.

A

⁢

X

→

Y

⁢

X

3.

Y

⁢

X

→

Y

⁢

D

4.

Y

⁢

D

→

C

⁢

D

where

X

,

Y

are new non-terminals introduced to

G

. Note also that, among the new forms, 1 and 3 are right context-sensitive, while 2 and 4 are left context-sensitive. Thus, a grammar in Kuroda normal form is equivalent to a grammar with productions having one of the following forms:

1.

A

→

a

,

2.

A

→

B

⁢

C

,

3.

A

⁢

B

→

A

⁢

C

,

4.

A

⁢

B

→

C

⁢

B

.

It can be shown that

Theorem 2

.

A grammar in Kuroda normal form if it is equivalent to a grammar whose productions are in one of forms

1

,

2

, or

3

.

By

symmetry

, a grammar in Kuroda normal form is equivalent to a grammar whose productions are in one of forms

1

,

2

, or

4

. A grammar whose productions are in one of forms

1

,

2

, or

3

is said to be in

one-sided normal form

.

As a

corollary

, every

λ

-free context-sensitive language (not containing the empty word

λ

) can be generated by a grammar in one-sided normal form.

What if we throw in production of the form

A

→

λ

in the above list? Then certainly every context-sensitive language has a grammar in this “extended” normal form. In fact, we have

Theorem 3

.

Every

type-0 language

can be generated by a grammar whose productions are in one of the following forms:

1.

A

→

a

,

2.

A

→

B

⁢

C

,

3.

A

⁢

B

→

A

⁢

C

,

4.

A

→

λ

.

References

1

G. E. Révész,

Introduction

to

Formal Languages

, Dover Publications (1991).

2

A. Mateescu, A. Salomaa,

Chapter 4 - Aspects of Classical

Language

Theory, Handbook of Formal Languages: Volume 1. Word, Language, Grammar

, Springer, (1997).

Title

Kuroda normal form

Canonical

name

KurodaNormalForm

Date of creation

2013-03-22 18:58:07

Last modified on

2013-03-22 18:58:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Classification

msc 68Q45

Related topic

GreibachNormalForm

Related topic

ChomskyNormalForm

Defines

one-sided normal form

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