A hierarchy of grammars

 

Type 0 : free or unrestricted grammars

These are the most general. Productions are of the form u �C> v where both u and v are arbitrary strings of symbols in V , with u non-null. There are no restrictions on what appears on the left or right-hand side other than the left-hand side must be non-empty.

Type 1 : context-sensitive grammars

Productions are of the form uXw �C> uvw where u , v and w are arbitrary strings of symbols in V , with v non-null, and X a single nonterminal. In other words, X may be

replaced by v but only when it is surrounded by u and w . (i.e. in a particular context).

 

Type 2 : context-free grammars

Productions are of the form X�C> v where v is an arbitrary string of symbols in V , and X is a single nonterminal. Wherever you find X, you can replace with v (regardless of context).

Type 3 : regular grammars

Productions are of the form X�C> a or X�C> aY where X and Y are nonterminals and a is a terminal. That is the left- hand side must be a single nonterminal and the right-hand side can be either a single terminal by itself or with a single nonterminal. These grammars are the most limited in terms of expressive power.

 

通过对产生式施加不同的限制， Chomsky 将文法 分为四种类型：

0 型文法：对任一产生式 α→β，都有 α∈(V N ∪V T ) + ， β∈(V N ∪V T ) *

1型文法： 对任一产生式 α→β，都有|β|≥|α|， 仅仅 S→ε除外

2型文法： 对任一产生式 α→β，都有α∈V N  ， β∈(V N ∪V T ) *

3型文法： 任一产生式 α→β的形式都为A→aB 或 A→a，其中A∈V N  ，B∈V N  ，a∈V T