The Chomsky hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages.  This hierarchy was described by Noam Chomsky in 1956.

Formal grammars

A formal grammar consists of a finite set of terminal symbols (the letters of the words in the formal language), a finite set of nonterminal symbols, a set of production rules with a left- and a right-hand side consisting of a word of these symbols, and a start symbol.  A rule may be applied to a word by replacing the left-hand side by the right-hand side.  A derivation is a sequence of rule applications.  Such a grammar defines the formal language of all words consisting solely of terminal symbols that can be reached by a derivation from the start symbol.

Nonterminals are usually represented by uppercase letters, terminals by lowercase letters, and the start symbol by an "S".  For example, the grammar with terminals {a, b}, nonterminals {S, A, B}, production rules

S -> ABS

S -> ε  (with ε the empty string)

BA -> AB

BS -> b

Bb -> bb

Ab -> ab

Aa -> aa

and start symbol S, defines the language of all words of the form aⁿbⁿ. (ie: n copies of a followed by n copies of b)(See formal grammar for a more elaborate explanation.)

The hierarchy

The Chomsky hierarchy consists of the following levels:

• Type-0 grammars (unrestricted grammars) include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine.  The language that is recognized by a Turing machine is defined as all the strings on which it halts. These languages are also known as the recursively enumerable languages.  Note that this is different from the [recursive languages]? which can be decided by an always halting Turing machine.

• Type-1 grammars (context-sensitive grammars) generate the context-sensitive languages.  These grammars have rules of the form αAβ -> αγβ with A a nonterminal and α, β and γ strings of terminals and nonterminals. The strings α and β may be empty, but γ must be nonempty.  It can also include the rule S->ε.  If it does, then it must not have an S on the right side of any rule.  These languages are exactly all languages that can be recognized by a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.

• Type-2 grammars (context-free grammars) generate the context-free languages. These are defined by rules of the form A -> γ with A a nonterminal and γ a string of terminals and nonterminals. These languages are exactly all languages that can be recognized by a nondeterministic pushdown automaton. Context free languages are the theoretical basis for the syntax of most programming languages.

• Type-3 grammars (regular 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. The rule S -> ε is also here allowed 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 used to define search patterns and the lexical structure of programming languages.

Every regular language is context-free, every context-free language is context-sensitive and every context-sensitive language is recursively enumerable. These are all proper inclusions, meaning that there exist recursively enumerable languages which are not context-sensitive, context-sensitive languages which are not context-free and context-free languages which are not regular.

The following table summarizes each of Chomsky's four types of grammars, the class of languages it generates, the type of automaton that recognizes it, and the form its rules must have. 

Grammar	Languages	Automaton	Production rules
Type-0	Recursively enumerable	Turing machine	No restrictions
Type-1	Context-sensitive	Linear-bounded nondeterministic Turing machine	αAβ -> αγβ
Type-2	Context-free	Nondeterministic pushdown automaton	A -> γ
Type-3	Regular	Finite state automaton	A -> aB A -> a

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References:

• Noam Chomsky: Three models for the description of language, IRE Transactions on Information Theory, 2 (1956), pages 113-124
• Noam Chomsky: On certain formal properties of grammars, Information and Control, 1 (1959), pages 91-112

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