Chomsky hierarchy

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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 Marcel-Paul Schützenberger who played a crucial role in the development of the theory of formal languages.



[edit] Formal grammars

A formal grammar of this type consists of:

A formal grammar defines (or generates) a formal language, which is a (usually infinite) set of finite-length 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 left-hand side of the rule with those that appear on the right-hand 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

S \rightarrow \, ABS
S \rightarrow \, ε (where ε is the empty string)
BA \rightarrow \, AB
BS \rightarrow \, b
Bb \rightarrow \, bb
Ab \rightarrow \, ab
Aa \rightarrow \, aa

and start symbol S, defines the language of all words of the form anbn (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

S \rightarrow \, aSb
S \rightarrow \, ε

[edit] The hierarchy

The Chomsky hierarchy.
Set inclusions described by the Chomsky hierarchy.

The Chomsky hierarchy consists of the following levels:

Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy.

Every regular language is context-free, every context-free language, not containing the empty string, is context-sensitive and 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 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 language it generates, the type of automaton that recognizes it, and the form its rules must have.

Grammar Languages Automaton Production rules (constraints)
Type-0 Recursively enumerable Turing machine \alpha \rightarrow \beta (no restrictions)
Type-1 Context-sensitive Linear-bounded non-deterministic Turing machine \alpha A\beta \rightarrow \alpha\gamma\beta
Type-2 Context-free Non-deterministic pushdown automaton A \rightarrow \gamma
Type-3 Regular Finite state automaton A \rightarrow a
A \rightarrow aB

However, there are further categories of formal languages, some of which are given in the following table:

[edit] See also

[edit] References

  1. ^ Chomsky, Noam (1956). "Three models for the description of language". IRE Transactions on Information Theory (2): 113–124. 

[edit] External links

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