Chomsky hierarchy is a hierarchical arrangement of classes of formal grammar. According to Chomsky's Hierarchy, grammar is of four types, which are as follows:
Type 0 grammar is unrestricted grammar and is recognized using a Turing machine.
Type 1 grammar is context-sensitive grammar and is recognized using a Linear Bounded Automata.
Type 2 grammar is context-free grammar and is recognized using a Push Down Automata.
Type 3 grammar is regular grammar and is recognized using a Finite Automata.
Type 0: Unrestricted Grammar:
Type-0 grammars include all formal grammar and are known as unrestricted grammar. Type 0 grammar languages are recognized by turing machine. The language accepted by this type of grammar are also known as the Recursively Enumerable languages. Type 0 grammar has a Production in the form of [α→β] where α is ( V + T)* V ( V + T)* V : Variables T : Terminals. β is ( V + T )*. Note: In type 0 there must be at least one variable on the Left side of production.
Type 1: Context-sensitive Grammar:
Type-1 grammars are context-sensitive grammars and generate context-sensitive language. Type 1 grammar languages are recognized by linear bounded automata. The language accepted by this type of grammar is context-sensitive language.
Type 1 grammar has a Production in the form of [α→β] where
α is ( V + T)* V ( V + T)*
V : Variables
T : Terminals.
β is ( V + T )+.
Note: In type 0 there must be at least one variable on the Left side of production.
Type 2: Context-free Grammar:
Type-2 grammars are context-free grammars and generate context-free language. Type 2 grammar languages are recognized by push-down automata. The language accepted by this type of grammar is context-free language. The left-hand side of production can have only one variable and there is no restriction on the right-hand side of production. That is,
[α]=1
Type 3: Regular Grammar:
Type-3 grammars generate regular languages. Regular languages are accepted by a finite-state automaton. Type 3 is the most restricted form of grammar. The grammar should be in the given form only
V →VT/T (left-regular grammar)
V→TV/T (right-regular grammar)
Where V is variable and T is terminal.
All the grammars are related to each other in such a way that, Regular grammar ⊃ Context-free Grammar ⊃ Context-sensitive grammar ⊃ Unrestricted Grammar.
Chomsky Hierarchy in TOC: https://www.geeksforgeeks.org/chomsky-hierarchy-in-theory-of-computation/