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



\alpha \to \beta

where

\alpha is ( V + T)* V ( V + T)*
V : Variables
T : Terminals.

\beta 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



\alpha \to \beta

|\alpha| <= |\beta|

i.e count of symbol in \alpha is less than or equal to \beta
 
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. Type-2 grammars generate the context-free languages.
In Type 2,
1. First of all it should be Type 1.
2. Left hand side of production can have only one variable.

|\alpha| = 1.

Their is no restriction on \beta.

 

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*.
(or)
V –> T*V /T*

 
for example :
S –> ab.

 

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

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