Chomsky hierarchy

The Chomsky hierarchy or Chomsky-Schützenberger hierarchy is a way of classifying formal grammars into four types, with the lower numbered types being more general.

Recall that a formal grammar $G=(\Sigma,N,P,\sigma)$ consists of an alphabet $\Sigma$, an alphabet $N$ of non-terminal symbols properly included in $\Sigma$, a non-empty finite set  $P$ of productions, and a symbol $\sigma\in N$ called the start symbol. The non-empty alphabet $T:=\Sigma-N$ is the set of terminal symbols. Then $G$ is called a

Type-0 grammar
Type-1 grammar

if the productions are of the form $uAv\to uWv$, where $u,v,W\in\Sigma^{*}$ with $W\neq\lambda$, and $A\in N$, or $\sigma\to\lambda$, provided that $\sigma$ does not occur on the right hand side of any productions in $P$. As $A$ is surrounded by words $u,v$, a type-1 grammar is also known as a context-sensitive grammar.

Type-2 grammar

if the productions are of the form $A\to W$, where $A\in N$ and $W\in\Sigma^{*}$. Type-2 grammars are also called context-free grammars, because the left hand side of any productions are “free” of contexts.

Type-3 grammar

if the productions are of the form $A\to u$ or $A\to uB$, where $A,B\in N$ and $u\in T^{*}$. Owing to the fact that languages  generated by type-3 grammars can be represented by regular expressions  , type-3 grammars are also known as regular grammars.

It is clear that every type-$i$ grammar is type-0, and every type-3 grammar is type-2. A type-2 grammar is not necessarily type-1, because it may contain both $\sigma\to\lambda$ and $A\to W$, where $\lambda$ occurs in $W$. Nevertheless, the relevance of the hierarchy has more to do with the languages generated by the grammars. Call a formal language a type-$i$ language if it is generated by a type-$i$ grammar, and denote $\mathscr{L}_{i}$ the family of type-$i$ languages. Then it can be shown that

 $\mathscr{L}_{3}\subset\mathscr{L}_{2}\subset\mathscr{L}_{1}\subset\mathscr{L}_% {0}$

where each inclusion is strict.

Below is a table summarizing the four types of grammars, the languages they generate, and the equivalent     computational devices accepting the languages.

 Title Chomsky hierarchy Canonical name ChomskyHierarchy Date of creation 2013-03-22 16:27:04 Last modified on 2013-03-22 16:27:04 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 13 Author CWoo (3771) Entry type Definition Classification msc 68Q15 Classification msc 03D55 Synonym Chomsky-Schützenberger hierarchy Synonym Chomsky-Schutzenberger hierarchy Synonym unrestricted grammar Defines type-0 grammar Defines type-0 language \@unrecurse