Chomsky Normal Form

A CFG is in Chomsky Normal Form if the Productions are in the following forms −

A → a

A → BC

S → ε

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

Algorithm to Convert into Chomsky Normal Form −

Step 1 − If the start symbol S occurs on some right side, create a new start symbol S’ and a new production S’→ S.

Step 2 − Remove Null productions. (Using the Null production removal algorithm discussed earper)

Step 3 − Remove unit productions. (Using the Unit production removal algorithm discussed earper)

Step 4 − Replace each production A → B₁…B_n where n > 2 with A → B₁C where C → B₂ …B_n. Repeat this step for all productions having two or more symbols in the right side.

Step 5 − If the right side of any production is in the form A → aB where a is a terminal and A, B are non-terminal, then the production is replaced by A → XB and X → a. Repeat this step for every production which is in the form A → aB.

Problem

Convert the following CFG into CNF

S → ASA | aB, A → B | S, B → b | ε

Solution

(1) Since S appears in R.H.S, we add a new state S₀ and S₀→S is added to the production set and it becomes −

S₀→S, S→ ASA | aB, A → B | S, B → b | ∈

(2) Now we will remove the null productions −

B → ∈ and A → ∈

After removing B → ε, the production set becomes −

S₀→S, S→ ASA | aB | a, A → B | S | ∈, B → b

After removing A → ∈, the production set becomes −

S₀→S, S→ ASA | aB | a | AS | SA | S, A → B | S, B → b

(3) Now we will remove the unit productions.

After removing S → S, the production set becomes −

S₀→S, S→ ASA | aB | a | AS | SA, A → B | S, B → b

After removing S₀→ S, the production set becomes −

S₀→ ASA | aB | a | AS | SA, S→ ASA | aB | a | AS | SA

A → B | S, B → b

After removing A→ B, the production set becomes −

S₀ → ASA | aB | a | AS | SA, S→ ASA | aB | a | AS | SA

A → S | b

B → b

After removing A→ S, the production set becomes −

S₀ → ASA | aB | a | AS | SA, S→ ASA | aB | a | AS | SA

A → b |ASA | aB | a | AS | SA, B → b

(4) Now we will find out more than two variables in the R.H.S

Here, S₀→ ASA, S → ASA, A→ ASA violates two Non-terminals in R.H.S.

Hence we will apply step 4 and step 5 to get the following final production set which is in CNF −

S₀→ AX | aB | a | AS | SA

S→ AX | aB | a | AS | SA

A → b |AX | aB | a | AS | SA

B → b

X → SA

(5) We have to change the productions S₀→ aB, S→ aB, A→ aB

And the final production set becomes −

S₀→ AX | YB | a | AS | SA

S→ AX | YB | a | AS | SA

A → b A → b |AX | YB | a | AS | SA

B → b

X → SA

Y → a