# Chomsky Hierarchy in Theory of Computation

According to Chomsky hierarchy, grammars are divided of 4 types:

Type 0 known as unrestricted grammar. Type 1 known as context sensitive grammar. Type 2 known as context free grammar. Type 3 Regular Grammar.

**Type 0: Unrestricted Grammar: **

In Type 0

Type-0 grammars include all formal grammars. Type 0 grammar language are recognized by turing machine. These languages are also known as the Recursively Enumerable languages.

Grammar Production in the form of

where

is ( V + T)* V ( V + T)*

V : Variables

T : Terminals.

is ( V + T )*.

In type 0 there must be at least one variable on Left side of production.

For example,

Sab –> ba

A –> S.

Here, Variables are S, A and Terminals a, b.

**Type 1: Context Sensitive Grammar)**

Type-1 grammars generate the context-sensitive languages. The language generated by the grammar are recognized by the Linear Bound Automata

In Type 1

I. First of all Type 1 grammar should be Type 0.

II. Grammar Production in the form of

|| <= ||

i.e count of symbol in is less than or equal to

For Example,

S –> AB

AB –> abc

B –> b

**Type 2: Context Free Grammar:**

Type-2 grammars generate the context-free languages. The language generated by the grammar is recognized by a Pushdown automata.

In Type 2,

1. First of all it should be Type 1.

2. Left hand side of production can have only one variable.

|| = 1.

Their is no restriction on .

For example,

S –> AB

A –> a

B –> b

**Type 3: Regular Grammar:**

Type-3 grammars generate regular languages. These languages are exactly all languages that can be accepted by a finite state automaton.

Type 3 is most restricted form of grammar.

Type 3 should be in the given form only :

**V –> VT / T ** (left-regular grammar)

**(or)**

**V –> TV /T ** (right-regular grammar)

for example:

S –> a

The above form is called as strictly regular grammar.

There is another form of regular grammar called extended regular grammar. In this form :

**V –> VT* / T*. **(extended left-regular grammar)**(or) ****V –> T*V /T*** (extended right-regular grammar)

for example :

S –> ab.

**REFERENCES**

https://en.wikipedia.org/wiki/Chomsky_hierarchy

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