- Automata Theory Tutorial
- Automata Theory - Home
- Automata Theory Introduction
- Deterministic Finite Automaton
- Non-deterministic Finite Automaton
- NDFA to DFA Conversion
- DFA Minimization
- Moore & Mealy Machines

- Classification of Grammars
- Introduction to Grammars
- Language Generated by Grammars
- Chomsky Grammar Classification

- Regular Grammar
- Regular Expressions
- Regular Sets
- Arden's Theorem
- Constructing FA from RE
- Pumping Lemma for Regular Grammar
- DFA Complement

- Context-Free Grammars
- Context-Free Grammar Introduction
- Ambiguity in Grammar
- CFL Closure Properties
- CFG Simplification
- Chomsky Normal Form
- Greibach Normal Form
- Pumping Lemma for CFG

- Pushdown Automata
- Pushdown Automata Introduction
- Pushdown Automata Acceptance
- PDA & Context Free Grammar
- PDA & Parsing

- Turing Machine
- Turing Machine Introduction
- Accepted & Decided Language
- Multi-tape Turing Machine
- Multi-Track Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- Linear Bounded Automata

- Decidability
- Language Decidability
- Undecidable Language
- Turing Machine Halting Problem
- Rice Theorem
- Post Correspondence Problem

- Automata Theory Useful Resources
- Automata Theory - Quick Guide
- Automata Theory - Useful Resources
- Automata Theory - Discussion

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.

**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 earlier)

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

**Step 4** − Replace each production **A → B _{1}…B_{n}** where

**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**.

Convert the following CFG into CNF

S → ASA | aB, A → B | S, B → b | ε

**(1)** Since **S** appears in R.H.S, we add a new state **S _{0}** and

S_{0}→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 −

S_{0}→S, S→ ASA | aB | a, A → B | S | ∈, B → b

After removing A → ∈, the production set becomes −

S_{0}→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 −

S_{0}→S, S→ ASA | aB | a | AS | SA, A → B | S, B → b

After removing S_{0}→ S, the production set becomes −

S_{0}→ ASA | aB | a | AS | SA, S→ ASA | aB | a | AS | SA

A → B | S, B → b

After removing A→ B, the production set becomes −

S_{0} → ASA | aB | a | AS | SA, S→ ASA | aB | a | AS | SA

A → S | b

B → b

After removing A→ S, the production set becomes −

S_{0} → 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, S_{0}→ 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 −

S_{0}→ 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 S_{0}→ aB, S→ aB, A→ aB

And the final production set becomes −

S_{0}→ 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

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