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Chomsky Hierarchy in Theory of Computation

  • Difficulty Level : Easy
  • Last Updated : 07 Oct, 2021

According to Chomsky hierarchy, grammars is divided into 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.

 

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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
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          (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 

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
 

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