A hierarchy of grammars
Type 0: free or unrestricted grammars
These are the most general. Productions are of the form u –> 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 –> 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–> 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–> a or X–> 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∈(VN∪VT)+， β∈(VN∪VT)*
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∈VN ， β∈(VN∪VT)*
3型文法：任一产生式α→β的形式都为A→aB或A→a，其中A∈VN ，B∈VN ，a∈VT
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