Chomsky hierarchy

A formal grammar of this type consists of a finite set of production rules (left-hand side → right-hand side), where each side consists of a finite sequence of the following symbols:

• a finite set of nonterminal symbols (indicating that some production rule can yet be applied)
• a finite set of terminal symbols (indicating that no production rule can be applied)
• a start symbol (a distinguished nonterminal symbol)

A formal grammar provides an axiom schema for (or generates) a formal language, which is a (usually infinite) set of finite-length sequences of symbols 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 by replacing an occurrence of the symbols on its left-hand side with those that appear on its 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 often 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 → AB

S → ε (where ε is the empty string)

A → aS

B → b

and start symbol S, defines the language of all words of the form ${\displaystyle a^{n}b^{n}}$ (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

S → aSb

S → ε

As another example, a grammar for a toy subset of the English language is given by:

terminals

{generate, hate, great, green, ideas, linguists}

nonterminals

{S, NP, VP, N, V, Adj}

production rules

S → NP VP

NP → Adj NP

NP → N

VP → V NP

VP → V

N → ideas

N → linguists

V → generate

V → hate

Adj → great

Adj → green

and start symbol S. An example derivation is

S → NP VP → Adj NP VP → Adj N VP → Adj N V NP → Adj N V Adj NP → Adj N V Adj Adj NP → Adj N V Adj Adj N → great N V Adj Adj N → great linguists V Adj Adj N → great linguists generate Adj Adj N → great linguists generate great Adj N → great linguists generate great green N → great linguists generate great green ideas.

Other sequences that can be derived from this grammar are: "ideas hate great linguists", and "ideas generate". While these sentences are nonsensical, they are syntactically correct. A syntactically incorrect sentence (e.g. "ideas ideas great hate") cannot be derived from this grammar. See "Colorless green ideas sleep furiously" for a similar example given by Chomsky in 1957; see Phrase structure grammar and Phrase structure rules for more natural language examples and the problems of formal grammar in that area.

Set inclusions described by the Chomsky hierarchy

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)*	Examples[3]
Type-0	Recursively enumerable	Turing machine	${\displaystyle \gamma \rightarrow \alpha }$ (no constraints)	${\displaystyle L=\{w|w}$ describes a terminating Turing machine${\displaystyle \}}$
Type-1	Context-sensitive	Linear-bounded non-deterministic Turing machine	${\displaystyle \alpha A\beta \rightarrow \alpha \gamma \beta }$	${\displaystyle L=\{a^{n}b^{n}c^{n}|n>0\}}$
Type-2	Context-free	Non-deterministic pushdown automaton	${\displaystyle A\rightarrow \alpha }$	${\displaystyle L=\{a^{n}b^{n}|n>0\}}$
Type-3	Regular	Finite state automaton	${\displaystyle A\rightarrow {\text{a}}}$ and ${\displaystyle A\rightarrow {\text{a}}B}$	${\displaystyle L=\{a^{n}|n\geq 0\}}$
* Meaning of symbols: ${\displaystyle {\text{a}}}$ = terminal ${\displaystyle A}$, ${\displaystyle B}$ = non-terminal ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \gamma }$ = string of terminals and/or non-terminals ${\displaystyle \alpha }$, ${\displaystyle \beta }$ = maybe empty ${\displaystyle \gamma }$ = never empty

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.[4]

• Chomsky, Noam (1959). "On certain formal properties of grammars" (PDF). Information and Control. 2 (2): 137–167. doi:10.1016/S0019-9958(59)90362-6.
• Chomsky, Noam; Schützenberger, Marcel P. (1963). "The algebraic theory of context free languages". In Braffort, P.; Hirschberg, D. (eds.). Computer Programming and Formal Systems (PDF). Amsterdam: North Holland. pp. 118–161.
• Davis, Martin D.; Sigal, Ron; Weyuker, Elaine J. (1994). Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science (2nd ed.). Boston: Academic Press, Harcourt, Brace. p. 327. ISBN 0-12-206382-1.