Language Is Cool. The Chomsky Hierarchy. Normal Forms. Chomsky Normal Form (CNF) & Greibach Normal Form (GNF) Teori Bahasa & Otomata - Danang Junaedi

Chomsky Normal Form (CNF) &

Greibach Normal Form (GNF)

Dosen Pembina Danang Junaedi

IF-UTAMA 2

Language Is Cool

• Language: A protocol for the transmission of concepts and intentions between humans

– Documentation is not available – Documentation does not really work – Learned through exposure and use

• Significant amount of internal structure, redundancy, and consistency

• Who makes language? – Kids.

• Adults coin words here and there, but when they’re forced to invent a common language to get things done, it’s called a Pidgin, and it’s terrible

• The kids hear it, and invent a Creole – a merged language of significantly greater accuracy and depth

• Children make languages • Adults make “working” languages

• Programmers make barely working languages

Normal Forms

• Normal forms are special types of context-free

languages.

• Having more restricted (but still powerful)

grammar forms make important algorithms

efficient.

• Two widely-known forms: Chomsky Normal and

Greibach Normal

The Chomsky Hierarchy

Type Language Grammar Automaton

0 Partially Computable

Unrestricted DTM - NTM

1 Context Sensitive Context Sensitive Linearly Bounded Automaton 2 Context Free Context Free NPDA 3 Regular right regular,

left regular DFA, NFA

0

Type

1

Type

2

Type

3

Type

⊂

⊂

⊂

You don’t have to know this

IF-UTAMA 5

The Chomsky Hierarchy

Context Free Languages Computable Languages Partially Computable Languages

{an_bn_{, n ≥ 0}} Regular Languages {an_bn_cn_{, n ≥ 0}} {am_bn_{, m,n ≥ 0}} {M, H(<M>)} IF-UTAMA 6

Chomsky Normal Form (CNF)

A CFG is in Chomsky Normal Form if all its productions

are of the form:

A

→

BC or

A

→

a

where A, B, C

∈

V and a

∈

T. Also, S

→ ε

may be one of

the productions.

Examples of CNF

Example 1:

S

→

AB

A

→

BC | CC | a

B

→

CB | b

C

→

c

Example 2:

S

→

AB |

BC | AC |

ε

A

→

BC | a

B

→

AC | b

C

→

AB | c

Examples of CNF

b

A

SA

A

a

S

AS

S

→

→

→

→

Not Chomsky

Normal Form

aa

A

SA

A

AAS

S

AS

S

→

→

→

→

Chomsky

Normal Form

IF-UTAMA 9

Is that all Context Free Grammars can be

expressed in Chomsky Normal Form?

Consider the following simple grammar:

A

→

cA | a

B

→

ABC | b

C

→

c

How to convert this grammar to CNF?

CNF

IF-UTAMA 10

Algorithm for Converting to CNF

•

Step (1)

Eliminate ε-productions and unit productions

•

Step (2)

For remaining production α → β

not form

A

→

BC

_{nor of form}

A

_→

a

_{, replace occurrences of}

terminals

a

_,

b

_,

c

_{, … in}

β

_{with new nonterminal}

representatives

C

_a

_,

C

_b

_,

C

_c

_{, … and then add new}

productions

C

_a

→

a C

_b

→

b C

_c

→

c

_…

•

Step (3)

If right-hand side of any production contains

three or more nonterminals, then decompose this

production into a series of productions the right-hand

sides of which consist of exactly two nonterminals.

Power of Chomsky Normal

• Every CFG can be rewritten in Chomsky Normal

Form

• Step 1: For any production rule with more than

one terminal on the right, substitute all with

variables.

• Step 2: Substitute more variables in order to

make the variable strings shorter.

Step 1: Convert every production into either:

A

→

B

1

B

2

…

B

n

or

A

→

a

e.g. A

→

bCDeF becomes:

A

→

BCDEF

B

→

b

E

→

e

Step 2: Convert production of the form A

→

B

1

B

2

…

B

n

into A

→

C

₁

C

₂

:

e.g. A

→

BCDEF becomes:

A

→

BX

X

→

CY

Y

→

DZ

Z

→

EF

Conversion into CNF

IF-UTAMA 13

Example: Convert to CNF

Original:

S -> ABa

A -> aab

B -> Ac

Step 1:

S -> ABB

_a

A -> B

_a

B

_a

B

_b

B -> AB

_c

B

_a

-> a

B

_b

-> b

B

_c

-> c

Step 2:

S -> AD

1

D

₁

->BB

_a

A -> B

a

D

2

D

2

->B

a

B

b

B -> AB

c

B

a

-> a

B

b

-> b

B

c

-> c

IF-UTAMA 14

Convert the following CFG into Chomsky

Normal Form:

S

→ ε

S

→

ABBA

B

→

bCb

A

→

a

C

→

c

Class Discussion

Greibach Normal Form (GNF)

A CFG is in Greibach Normal Form if all its productions

are of the form:

A

→

a

α

where A

∈

V, a

∈

T and

α ∈

V*. Also, S

→ ε

may be

one of the productions.

Advantages:

• Every derivation of a string

s

_{contains |}

s

_{| rule}

applications.

• Greibach normal form grammars can easily be

converted to pushdown automata with no

ε

-transitions.

This is useful because such PDAs are guaranteed to

halt.

Examples of GNF

Example 1: S

→

aABC

A

→

aA | a

B

→

bB | b

C

→

cC | c

Example 2: S

→

bAB |

ε

A

→

aBAA | aAAB | a

B

→

bABB | bBBA | b

IF-UTAMA 17

Conversion Algorithm

•

Step (1)

Find equivalent grammar

G´

_{in CNF}

•

Step (2)

Order nonterminals of

G´

_from

X

₁

_to

X

_n

_.

•

Step (3)

Work upward through nonterminals of

G´

_,

making replacements so as to

ensure that all

ultimately ascending.

•

Step (4)

Work downward through nonterminals,

making replacements so as to ensure that all

productions ultimately in GNF, i.e., of form

X

→

a

α,

where

X

_nonterminal,

a

is a terminal, and α

_(possibly

empty) string of non-terminals.

IF-UTAMA 18

Is that all Context Free Grammars can be

expressed in Greibach Normal Form?

Transformation can be done by a

combination of substitution and removal

of left recursion.

GNF

Removal of Left Recursion

We want to remove all the productions in the form of:

A

→

A

α

This is called a left recursion.

Replace:

X

→ α

₁

|

α

₂

|

…

|

α

_n

X

→

X

β

1

| X

β

2

|

…

| X

β

n

by:

X

→ α

1

|

α

2

|

…

|

α

n

X

→ α

1

Z |

α

2

Z |

…

|

α

n

Z

Z

→ β

1

|

β

2

|

…

|

β

n

Z

→ β

1

Z |

β

2

Z |

…

|

β

n

Z

where X

∈

V

α

_i

,

β

_i

∈

(V

∪

T)*

such that

α

i

does

not start with X

Tahapan Penghilangan Rekursif Kiri

[5]

1. Pisahkan aturan produksi yang rekursif kiri dan tidak rekursif kiri – Aturan produksi yang rekursif kiri

X →Xβ1| Xβ2| …| Xβn Kita dapatkan simbolβ1, β2,…,βn – Aturan produksi yang tidak rekursif kiri

X → α1| α2| …| αn

Kita dapatkan simbolα1, α2,…,αn

2. Lakukan penggantian aturan produksi yang rekursif kiri menjadi X → α1| α2| …| αn

X → α1Z | α2Z | …| αnZ Z → β1| β2| …| βn Z → β1Z | β2Z | …| βnZ

dimana simbol Z adalah variabel non terminal baru yang kita buat sesuai banyaknya variablel pada aturan produksi yang rekursif kiri

IF-UTAMA 21 Hilangkan rekursif kiri untuk aturan produksi berikut:

E E+T E T

Solusi:

1. Pisahkan aturan produksi yang rekursif kiri dan tidak rekursif kiri – Aturan produksi yang rekursif kiri

E E+T

Kita dapatkan simbolα₁ +T – Aturan produksi yang tidak rekursif kiri

E T

Kita dapatkan simbolβ₁ T

2. Lakukan penggantian aturan produksi yang rekursif kiri menjadi E →T

E →TZ Z →+T Z →+TZ

Contoh Penghilangan Rekursif Kiri

Aturan produksi baru yang dihasilkan dari pengilangan aturan produksi yang rekursif kiri

IF-UTAMA 22

Studi Kasus

[5]

Hilangkan rekursif kiri dari aturan-aturan

produksi berikut ini:

1. S

Sab | aSc | dd | ff | Sbd

2. S

Sab | Sb | cA

A

Aa | a | bd

3. S

Sa | aAc | c |

ε

_A

_{Ab | ba}

Pembentukan GNF (1)

[5]

1. Aturan Produksi telah dalam bentuk CNF dan tidak menghasilkanε

2. Tentukan urutan simbol-simbol variabel yang ada dalam tata bahasa. Misalkan terdapat “n” variabel non terminal dengan urutanα₁, α₂, …,α_n

3. Berdasarkan urutan simbol tersebut seluruh aturan produksi dapat ditulis dalam bentuk

α_h α_i λ(dimana h ≤i)

a. Jika h < I, aturan produksi sudah benar

b. Jika h>I, lakukan subtitusi berulang (ganti α_I dengan variabel yang ada di ruas kanannya) sampai diperoleh bentuk

α_h α_p λ(dimana h ≤p)

• Jika h<p, aturan produksi sudah benar

• Jika h=p, lakukan penghilangan rekursif kiri

sisipkan simbol baru hasil penghilangan rekrusif kiri di posisi setelah simbolα_h

Pembentukan GNF dengan Subtitusi(2)

[5]

Jika step 2 dan 3 sudah dikerjakan akan menghasilkan aturan produksi dengan urutan yang benar yaituα_hα_iλ

(h<i) serta aturan produksi yang lain α_h a_λ (a simbol terminal) danβ_xλ(simbol baru dari hasil penghilangan rekursif kiri)

4. Bentuk normal greibach diperoleh dengan cara melakukan subtitusi mundur untuk simbolαmulai dariα_n sampaiα₁begitu juga untuk variabelβ

Contoh:

Diketahui Grammar G=(Vn,Vt,S,P), dimana Vn: {S,A,B,C,D}

V_t: {a,b,d} S: {S}

P: { SCA Aa|d Bb CDD DAB }

IF-UTAMA 25

Pembentukan GNF dengan Subtitusi(3)

[5]

Solusi:

2. Tentukan urutan simbol-simbol variabel yang ada dalam tata bahasa. Misalkan terdapat “n” variabel non terminal dengan urutanα₁, α₂, …,α_n

Aturan produksi menjadi

α₁α₄α₂ α₂a | d α₃b

α₄α₅α₅ α₅α₂α₃

3. Cek apakah berdasarkan urutan simbol tersebut seluruh aturan produksi dalam bentukα_h α_i λdimana (h≤i)

α1α4α2 (sudah benar karena 1<4)

α₄α₅α₅ (sudah benar karena 4<5)

α5 α2α3 (salah karena 5>2) =>subtitusi berulang (ganti α2 dengan

variabel yang ada di ruas kanannya) sampai diperoleh bentukα_h α_p λ

(dimana h ≤p) 5 4 3 2 1 } { : α α α α α ↓ ↓ ↓ ↓ ↓ D C B A S V_n IF-UTAMA 26

Pembentukan GNF dengan Subtitusi(4)

[5]

α5α2α3menjadiα5aα3| dα3

sehingga menghasilkan aturan produksi dengan urutan yang benar yaitu: α₁α₄α₂ α₂a | d α₃b α₄α₅α₅ α₅aα₃| dα₃ 4. Subtitusi mundur α₅aα₃| dα₃ α₄aα₃α₅ | dα₃α₅ α₁aα₃α₅α₂ | dα₃α₅ α₂

Ubah variabelα₁,…, α_nke variabel semula

Menjadi SaBDA | dBDA Aa | d B b C aBD | dBD D aB | dB D C B A S ↓ ↓ ↓ ↓ ↓ 5 4 3 2 1 α α α α α

Ubah aturan produksi berikut ini ke dalam

bentuk normal greibach:

1. S

→

c | Sa | SbA A

→

c

2. S

AA |d

A

SS | b

3. A

ABC

B

CA | b

C

AB | a

4. S

aSb | ab | SS

Studi Kasus

References

1. -,”Simplification of CFG”, http://www.geocities. com/ parouyr_sevak/03L11.ppt,Tanggal Akses 16 Mei 2009 2. Dr. William Thacker’s ,”Context-Free Grammars: Chapter 11

Normal Formal ”,http://cba.winthrop.

edu/thackerw/CSCI371/powerpoint%20slides/Ch11bNormalFor ms.ppt, Tanggal Akses 16 Mei 2009

3. -,” Context-Free Languages and Pushdown-Stack Automata : An Intermediate-Strength Model of Computation”,

http://home.manhattan.edu/~gregory.taylor/thcomp/slides/chapt er10.ppt, Tanggal Akses 16 Mei 2009

4. -,”Chomsky Hierarchy

Language Operations and Properties”,

http://www-cs.ccny.cuny.edu/~vmitsou/extra/Chomskyhier.ppt, Tanggal Akses 16 Mei 2009

5. Firrar Utdirartatmo, “Teori Bahasa dan Otomata”, Graha Ilmu, Yogyakarta, 2005