Cs133 lec05

Chomsky Normal Form

CS 133 : Automata Theory and Computability Context-Free Grammars and Languages Henry N. Adorna Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman {hnadorna,nshernandez}@dcs.upd.edu.ph

LEC 5

Chomsky Normal Form

Context-Free Grammars and Languages

Chomsky Normal Form

Chomsky Normal Form

Context-Free Grammars and Languages

Chomsky Normal Form

Chomsky Normal Form

Definition A CFG is in Chomsky normal form if every rule is of the form A A

→ →

BC a

where a is any terminal and A, B, C are any variables except that B and C may not be the start variable.

In addition we permit the rule S → ε, where S is the start variable.

Chomsky Normal Form

All CFGs can be converted to CNF.

Chomsky Normal Form

All CFGs can be converted to CNF.

Outline of conversion algorithm 1. Create a new start symbol S0 , with new rule S0 → S 2. Remove "nullable" variables 3. Remove "unit pairs" 4. Restructure rules whose right hand sides have length greater than 1 and include terminals 5. Restructure rules with long right hand sides

Chomsky Normal Form

All CFGs can be converted to CNF.

Outline of conversion algorithm 1. Create a new start symbol S0 , with new rule S0 → S 2. Remove "nullable" variables 3. Remove "unit pairs" 4. Restructure rules whose right hand sides have length greater than 1 and include terminals 5. Restructure rules with long right hand sides

Example S → ASA | aB A→ B |S B→ b|ε

Chomsky Normal Form

Convert the following CFG into CNF: 1. S A B

→ → →

ASB | ε aAS |a SbS | A |bb

2. S A B C

→ → → →

0A0 | 1B1 |BB C S|A S|ε

3. S A B

→ → →

AAA | B aA |B ε

4. S A B A B

→ → → → →

aAa | bBb | ε C |a C |b CDE |ε A | B |ab

Chomsky Normal Form

Questions?

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