The famous linguistic Noam Chomsky attempted to formalize the notion of grammar and languages in the 1950s. This effort, due to Chomsky, resulted in the definition of the "Chomsky Hierarchy", a hierarchy of language classes defined by gradually increasing the restrictions on the form of the productions. Chomsky numbered the four families of grammars (and languages) that make up the hierarchy and are defined as below.
Let G = ( N ,
, P, S ) be a grammar
-
G is called a Type-0 or unrestricted, or semi-there or phrase-structure grammar if all productions are of the form
, where 
and 
. -
G is a Type-1 or context-sensitive grammar if each production in P satisfies
such that and 
. Type-1 grammar , by special dispensation , is allowed to have the production 
, provided S does not appear on the right-hand side of any production. -
G is a Type-2 or context-free grammar if each production in P satisfies
i.e. 
is a single nonterminal. -
G is a Type-3 or right-linear or regular grammar if each production has one of the following three forms:
, A
b, A
where A, C N ( with A = C allowed ) and 
.