Within the field of computer science, specifically in the area of formal languages, the Chomsky hierarchy (occasionally referred to as Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars.
This hierarchy of grammars was described by Noam Chomsky in 1956.^{[1]} It is also named after MarcelPaul Schützenberger who played a crucial role in the development of the theory of formal languages.
Contents
Formal grammars
A formal grammar of this type consists of:
 a finite set of terminal symbols
 a finite set of nonterminal symbols
 a finite set of production rules with a left and a righthand side consisting of a sequence of these symbols
 a start symbol
A formal grammar defines (or generates) a formal language, which is a (usually infinite) set of finitelength sequences of symbols (i.e. strings) that may be constructed by applying production rules to another sequence of symbols which initially contains just the start symbol. A rule may be applied to a sequence of symbols by replacing an occurrence of the symbols on the lefthand side of the rule with those that appear on the righthand side. A sequence of rule applications is called a derivation. Such a grammar defines the formal language: all words consisting solely of terminal symbols which 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 S. For example, the grammar with terminals {a,b}, nonterminals {S,A,B}, production rules
and start symbol S, defines the language of all words of the form a^{n}b^{n} (i.e. n copies of a followed by n copies of b). The following is a simpler grammar that defines the same language: Terminals {a,b}, Nonterminals {S}, Start symbol S, Production rules
The hierarchy
The Chomsky hierarchy consists of the following levels:
 Type0 grammars (unrestricted grammars) include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine. 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 alwayshalting Turing machine.
 Type1 grammars (contextsensitive grammars) generate the contextsensitive languages. These grammars have rules of the form 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. The rule 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.)
 Type2 grammars (contextfree grammars) generate the contextfree languages. These are defined by rules of the form 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. Contextfree languages are the theoretical basis for the syntax of most programming languages.
 Type3 grammars (regular grammars) generate the regular languages. Such a grammar restricts its rules to a single nonterminal on the lefthand side and a righthand side consisting of a single terminal, possibly followed (or preceded, but not both in the same grammar) by a single nonterminal. The rule 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.
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