Explain Chomsky hierarchy in TOC

Chomsky Hierarchy represents the class of languages that are accepted by the different machines.

Chomsky hierarchy

Hierarchy of grammars according to Chomsky is explained below as per the grammar types −

Type 0 − It is an Unrestricted grammars

Unrestricted grammar − an unrestricted grammar is a 4-tuple (T,N,P,S), which consisting of −

T = set of terminals

N = set of nonterminal

P =as set of productions, of the form −


where v and w are strings consisting of nonterminal and terminals.

S = is called the start symbol.

Example − Turing Machine (TM)

Type 1 − Context-sensitive grammars

All productions are of form,

v -> w where |v| < |w|

uAv -> uwv with w != epsilon,

I.e., A -> w but only in the context of u _ v.

A context-sensitive grammar is equivalent to a linear bounded and context-sensitive language.

Example  − Linear Bounded Automaton (LBA)

Type 3 − Context-free grammars −

All productions are of the form −

A -> x − where A is nonterminal, x is a string of nonterminal and terminals, A context-free grammar is equivalent to a pushdown automaton (PDA) and to context free languages.

Example − Pushdown Automaton (PDA)

Type 3 − Regular grammars

All productions are of the form −

A -> xB

A -> x where A, B are nonterminal, x is a string of terminals.

Regular grammars are equivalent to regular sets and to finite automata.

Example  − Finite Automaton (FA)

Grammar typeGrammar acceptedLanguage acceptedAutomaton
Type 0unrestricted grammarrecursively enumerable languageTuring Machine
Type 1context-sensitive grammarcontext-sensitive languagelinear-bounded automata
Type 2Context-free grammarContext-free languagePush down automata
Type 3regular grammarregular languagefinite state automaton

The Chomsky hierarchy is depicted in the diagram given below −