The following table summarizes each of Chomsky's four types of grammars, the class of language it generates, the type of automaton that recognizes it, and the form its rules must have. The classes are defined by the constraints on the
productions rules. Note that the set of grammars corresponding to
recursive languages is not a member of this hierarchy; these would be properly between Type-0 and Type-1. Every regular language is context-free, every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable. These are all proper inclusions, meaning that there exist recursively enumerable languages that are not context-sensitive, context-sensitive languages that are not context-free and context-free languages that are not regular.
Regular (Type-3) grammars Type-3 grammars generate the
regular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal, in which case the grammar is
right regular. Alternatively, all the rules can have their right-hand sides consist of a single terminal, possibly
preceded by a single nonterminal (
left regular). These generate the same languages. However, if left-regular rules and right-regular rules are combined, the language need no longer be regular. The rule S \rightarrow \varepsilon is also allowed here if S does not appear on the right side of any rule. These languages are exactly all languages that can be decided by a
finite-state automaton. Additionally, this family of formal languages can be obtained by
regular expressions. Regular languages are commonly used to define search patterns and the lexical structure of programming languages. For example, the regular language L = \{a^n \mid n > 0\} is generated by the Type-3 grammar G = (\{S\}, \{a\}, P, S) with the productions P being the following. : : In
linguistics, a language that is not regular is called
supra-regular.
Context-free (Type-2) grammars Type-2 grammars generate the
context-free languages. These are defined by rules of the form A \rightarrow \alpha with A being a nonterminal and \alpha being a string of terminals and/or nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic
pushdown automaton. Context-free languages—or rather its subset of
deterministic context-free languages—are the theoretical basis for the phrase structure of most
programming languages, though their semantic analysis includes context-sensitive
name resolution due to declarations and
scope. Often a subset of grammars is used to make parsing easier, such as by an
LL parser. For example, the context-free language L = \{a^nb^n \mid n > 0\} is generated by the Type-2 grammar G = (\{S\}, \{a, b\}, P, S) with the productions P being the following. : : The language is context-free but not regular (by the
pumping lemma for regular languages). Every context-free language can be generated by a grammar in
Chomsky normal form.
Context-sensitive (Type-1) grammars Type-1 grammars generate
context-sensitive languages. These grammars have rules of the form \alpha A\beta \rightarrow \alpha\gamma\beta with A a nonterminal and \alpha, \beta and \gamma strings of terminals and/or nonterminals. The strings \alpha and \beta may be empty, but \gamma must be nonempty. The rule S \rightarrow \epsilon is allowed if S does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by a
linear bounded automaton (a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.) For example, the context-sensitive language L = \{a^nb^nc^n \mid n > 0\} is generated by the Type-1 grammar G = (\{S,A,B,C,W,Z\}, \{a, b, c\}, P, S) with the productions P being the following. : : : : : : : : : : The language is context-sensitive but not context-free (by the
pumping lemma for context-free languages). A proof that this grammar generates L = \{a^nb^nc^n \mid n > 0\} is sketched in the article on
Context-sensitive grammars.
Recursively enumerable (Type-0) grammars Type-0 grammars include all formal grammars. There are no constraints on the productions rules. They generate exactly all languages that can be recognized by a
Turing machine, thus any language that is possible to be generated can be generated by a Type-0 grammar. Note that this is different from the
recursive languages, which can be
decided by an
always-halting Turing machine. == Natural languages ==