Tag: Normal Forms in Automata Theory

Chomsky Normal Form | Normal Forms in Automata

Theory of Automata & Computation

Normal Forms-

 

Normalization is performed in order to standardize the grammar.

 

  • By reducing the grammar, the grammar gets minimized but does not gets standardized.
  • This is because the RHS of productions have no specific format.
  • In order to standardize the grammar, normalization is performed using normal forms.

 

Types of Normal Forms-

 

The most frequently used normal forms are-

 

 

  1. Chomsky Normal Form (CNF)
  2. Greibach Normal Form (GNF)

 

In this article, we will discuss about Chomsky Normal Form.

 

Chomsky Normal Form-

 

A context free grammar is said to be in chomsky normal form (CNF) if all its productions are of the form-

A → BC or A → a

where A, B, C are non-terminals and a is a terminal.

 

From here, we infer-

  • To be in CNF, all the productions must derive either two non-terminals or a single terminal.
  • CNF restricts the number of symbols on the right side of a production to be two.
  • The two symbols must be non-terminals or a single terminal.

 

Example-

 

S → AB

A → a

B → b

 

This context free grammar is in chomsky normal form.

 

Steps-

 

The following steps are followed to standardize the grammar using CNF-

 

Rule-01:

 

Reduce the grammar completely by-

  • Eliminating ∈ productions
  • Eliminating unit productions
  • Eliminating useless productions

 

Also Watch- How To Reduce Grammar?

 

Rule-02:

 

Replace each production of the form A → B1B2B3….Bn where n > 2 with A → B1C where C → B2B3….Bn.

Repeat this step for all the productions having more than two variables on RHS.

 

Rule-03:

 

Replace each production of the form A → aB with A → XB and X → a.

Repeat this step for all the productions having the form A → aB.

 

PRACTICE PROBLEMS BASED ON CHOMSKY NORMAL FORM-

 

Problem-01:

 

Convert the given grammar to CNF-

S → aAD

A → aB / bAB

B → b

D → d

 

Solution-

 

Step-01:

 

The given grammar is already completely reduced.

 

Step-02:

 

The productions already in chomsky normal form are-

B → b        ………..(1)

D → d        ………..(2)

These productions will remain as they are.

 

The productions not in chomsky normal form are-

S → aAD                ………..(3)

A → aB / bAB         ………..(4)

We will convert these productions in chomsky normal form.

 

Step-03:

 

Replace the terminal symbols a and b by new variables Ca and Cb.

 

This is done by introducing the following two new productions in the grammar-

Ca → a       ………..(5)

Cb → b       ………..(6)

 

Now, the productions (3) and (4) modifies to-

S → CaAD                  ………..(7)

A → CaB / CbAB         ………..(8)

 

Step-04:

 

Replace AD and AB by new variables CAD and CAB respectively.

 

This is done by introducing the following two new productions in the grammar-

CAD → AD       ………..(9)

CAB → AB       ………..(10)

 

Now, the productions (7) and (8) modifies to-

S → CaCAD                  ………..(11)

A → CaB / CbCAB         ………..(12)

 

Step-05:

 

From (1), (2), (5), (6), (9), (10), (11) and (12), the resultant grammar is-

 

S → CaCAD

A → CaB / CbCAB

B → b

D → d

Ca → a

Cb → b

CAD → AD

CAB → AB

 

This grammar is in chomsky normal form.

 

Problem-02:

 

Convert the given grammar to CNF-

S → 1A / 0B

A → 1AA / 0S / 0

B → 0BB / 1S / 1

 

Solution-

 

Step-01:

 

The given grammar is already completely reduced.

 

Step-02:

 

The productions already in chomsky normal form are-

A → 0        ………..(1)

B → 1        ………..(2)

These productions will remain as they are.

 

The productions not in chomsky normal form are-

S → 1A / 0B           ………..(3)

A → 1AA / 0S         ………..(4)

B → 0BB / 1S         ………..(5)

We will convert these productions in chomsky normal form.

 

Step-03:

 

Replace the terminal symbols 0 and 1 by new variables C and D.

 

This is done by introducing the following two new productions in the grammar-

C → 0       ………..(6)

D → 1       ………..(7)

 

Now, the productions (3), (4) and (5) modifies to-

S → DA / CB           ………..(8)

A → DAA / CS         ………..(9)

B → CBB / DS         ………..(10)

 

Step-04:

 

Out of (8), (9) and (10), the productions already in Chomsky Normal Form are-

S → DA / CB           ………..(11)

A → CS                   ………..(12)

B → DS                   ………..(13)

These productions will remain as they are.

 

The productions not in chomsky normal form are-

A → DAA         ………..(14)

B → CBB         ………..(15)

We will convert these productions in Chomsky Normal Form.

 

Step-05:

 

Replace AA and BB by new variables E and F respectively.

 

This is done by introducing the following two new productions in the grammar-

E → AA       ………..(16)

F → BB       ………..(17)

 

Now, the productions (14) and (15) modifies to-

A → DE         ………..(18)

B → CF         ………..(19)

 

Step-06:

 

From (1), (2), (6), (7), (11), (12), (13), (16), (17), (18) and (19), the resultant grammar is-

 

S → DA / CB

A → CS / DE / 0

B → DS / CF / 1

C → 0

D → 1

E → AA

F → BB

 

This grammar is in chomsky normal form.

 

To gain better understanding about Chomsky Normal Form,

Watch this Video Lecture

 

Next Article- Algorithm To Decide Whether CFL Is Empty

 

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