MELJUN CORTES Automata Lecture Chomsky Normal Form 2

8/21/2019 MELJUN CORTES Automata Lecture Chomsky Normal Form 2

1/16

Theory of Computation (With Automata Theory)

* Property of STI

Page 1 of 16

Chomsky Normal Form

CHOMSKY NORMAL

FORM

Chomsky Normal Form

Introduction

• Simplified versions of context-

free grammars are very useful

in analyzing the grammar,

constructing proofs about

certain properties of thegrammar, testing if a particular

string is a member of the

language of the grammar, etc.

• These simplified versions arecalled normal fo rms .

• One popular normal form of

context-free grammars is the

Chomsky Normal Form.


2/16


* Property of STI

Page 2 of 16

Chomsky Normal Form

Formal Definition of the ChomskyNormal Form

• A context-free grammar is in the

Chom sky norm al fo rm if every

rule is of the form

A → BC

A → a

where:

1. A, B, and C are any variables

2. a is any terminal

3. B and C may not be the start

variable

4. the rule S , where S is the

start variable, is allowed


3/16


* Property of STI

Page 3 of 16

Chomsky Normal Form

• Example:

Grammar G1:

S → XY 0

X → YY 1

Y → 0

Grammar G2:

S → XS XYY 0X 0

X → SX 11

Y → 1

Grammar G1 is in the

Chomsky normal form while

grammar G2 is not.


4/16


* Property of STI

Page 4 of 16

Chomsky Normal Form

• Procedure for converting acontext-free grammar into the

Chomsky Normal Form:

Case Study: Convert the

following grammar G3 into theChomsky normal form.

S ASA 0B

A B S

B 1

Step 1:

Create a new start variable S 0and the production S 0 S ,

where S is the original start

variable of the grammar being

converted.


5/16


* Property of STI

Page 5 of 16

Chomsky Normal Form

S0 S

S ASA 0B

A B S

B 1

Step 2:

For every variable A that is not

a start variable, remove all Arules (the -p roduc tions ).

Then, for each occurrence of A

on right hand side of a

production, add a new rulewith the A deleted.

For example, if there is a rule

B → AC , add the rule B → C .


6/16


* Property of STI

Page 6 of 16

Chomsky Normal Form

If there is a rule B → A, addthe rule B → ε, unless B → ε

was previously removed.

Removing B :

S0 S

S ASA 0B 0

A B S

B 1

Removing A :

S0

S

S ASA 0B 0SA AS S

A B S

B 1


7/16


* Property of STI

Page 7 of 16

Chomsky Normal Form

Step 3:

Remove all rules of the form A

B (the un it p roduc tions ).

For each rule B → w (where w

is a string of variables and

terminals) that appears, add

the rule A → w unless A → w

was previously removed.

Rules of the form A A can just simply be removed since

these are useless productions.


8/16


* Property of STI

Page 8 of 16

Chomsky Normal Form

Removing S S :

S0 S

S ASA 0B 0SA AS

A B S

B 1

Removing A B:

S0 S

S ASA 0B 0SA AS

A S 1

B 1


9/16


* Property of STI

Page 9 of 16

Chomsky Normal Form

Removing A S :

S0 S

S ASA 0B 0SA AS

A 1 ASA 0B 0SA AS

B 1

Removing S S 0:

S0 ASA 0B 0SA AS

S ASA 0B 0SA AS

A 1 ASA 0B 0SA AS

B 1


10/16


* Property of STI

Page 10 of 16

Chomsky Normal Form

Step 4:

Convert all the rules into the

proper form by removing the

rules of the form:

A w1w2…wk

where k ≥ 3 and w i is a

variable or terminal

and adding the rules A

w 1T 1, T 1 w 2T 2, T 2 w 3T 3,

and so on and so forth (where

T 1, T 2, and T 3 are new

variables).

If w i is a terminal, replace it

with a new variable T i and add

the rule T i w i .


11/16


* Property of STI

Page 11 of 16

Chomsky Normal Form

The resulting grammar G3 in

Chomsky normal form will be

S0 AT1 T2B 0 SA AS

S AT1 T2B 0 SA AS

A 1 AT1 T2B 0 SA AS

T1 SA

T2 0

B 1


12/16


* Property of STI

Page 12 of 16

Chomsky Normal Form

• Example:

Convert the following grammar

G4 into the Chomsky normal

form.

A BAB B

B 00

Step 1: Add S 0 A

S0 A

A BAB B

B 00


13/16


* Property of STI

Page 13 of 16

Chomsky Normal Form

Step 2: Remove -productions

Remove A :

S0 A

A BAB B BB

B 00

Remove B :

S0 A

A BAB B BB AB BA

B 00


14/16


* Property of STI

Page 14 of 16

Chomsky Normal Form

Step 3: Remove unit-productions

Remove A B:

S0 A

A BAB BB AB

BA 00

B 00

Remove S 0 A:

S0 BAB BB AB

BA 00

A BAB BB AB

BA 00

B 00


15/16


* Property of STI

Page 15 of 16

Chomsky Normal Form

Step 4: Convert all the rules into

the proper form.

The resulting grammar G4 in

Chomsky normal form will be:

S0 BT1 BB AB BA T2T2

A BT1 BB AB BA T2T2

B T2T2

T1 AB

T2 0


16/16


* Property of STI

Page 16 of 16

Chomsky Normal Form

• Formal Theorem on ChomskyNormal Form of Context-Free

Grammars:

Theo rem 6 :

Any con tex t-free grammar

can be converted in to i ts

Chomsky normal fo rm .

• Exercise

Convert the following context-

free grammar into its Chomsky

normal form:

S → ASB

A → 0AS 0

B → S1S A 00

MELJUN CORTES Automata Lecture Chomsky Normal Form 2

Documents