Within the field of computer science, specifically in the area of formal languages, the Chomsky
hierarchy is a containment hierarchy of classes of formal grammars. This hierarchy of grammars
was described by Noam Chomsky in 1956. It is also named after Marcel-Paul Schützenberger,
who played a crucial role in the development of the theory of formal languages. The Chomsky
Hierarchy, in essence, allows the possibility for the understanding and use of a computer
science model which enables a programmer to accomplish meaningful linguistic goals systematically.
Formal grammars
A formal grammar of this type consists of a finite set of production rules, where each
side consists of a sequence of the following symbols:
a finite set of nonterminal symbols a finite set of terminal symbols
a start symbol A formal grammar defines a formal language,
which is a 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 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 often represented by uppercase
letters, terminals by lowercase letters, and the start symbol by . For example, the grammar
with terminals , nonterminals , production rules
ε
and start symbol , defines the language of all words of the form . The following is a
simpler grammar that defines the same language: Terminals , Nonterminals , Start symbol , Production
rules
ε As another example, a grammar for a toy subset
of English language is given by terminals , nonterminals , production rules
and start symbol . An example derivation is SENTENCE NOUNPHRASE VERBPHRASE ADJ NOUNPHRASE
VERBPHRASE ADJ NOUN VERBPHRASE ADJ NOUN VERB NOUNPHRASE ADJ NOUN VERB ADJ NOUNPHRASE ADJ
NOUN VERB ADJ ADJ NOUNPHRASE ADJ NOUN VERB ADJ ADJ NOUN great NOUN VERB ADJ ADJ NOUN
great linguists VERB ADJ ADJ NOUN great linguists generate ADJ ADJ NOUN great linguists generate
great ADJ NOUN great linguists generate great green NOUN 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 like 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 grammars
in that area. The hierarchy
The Chomsky hierarchy consists of the following levels:
Type-0 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 always-halting Turing machine. Type-1 grammars generate the context-sensitive
languages. These grammars have rules of the form with a nonterminal and , and strings
of terminals and/or nonterminals. The strings and may be empty, but must be nonempty. The
rule is allowed if 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 Type-2 grammars generate the context-free
languages. These are defined by rules of the form with a nonterminal and 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 the subset of deterministic
context-free language – are the theoretical basis for the phrase structure of most programming
languages, though their syntax also includes context-sensitive name resolution due to declarations
and scope. Often a subset of grammars are used to make parsing easier, such as by an
LL parser. 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. Alternatively,
the right-hand side of the grammar can consist of a single terminal, possibly preceded by
a single nonterminal; 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
is also allowed here if 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.
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 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.
However, there are further categories of formal languages, some of which are given in the
expandable navigation box at the bottom of this page.
References
Chomsky, Noam. "On certain formal properties of grammars". Information and Control 2: 137–167.
doi:10.1016/S0019-9958(59)90362-6. Chomsky, Noam; Schützenberger, Marcel P..
"The algebraic theory of context free languages". In Braffort, P.; Hirschberg, D. Computer Programming
and Formal Languages. Amsterdam: North Holland. pp. 118–161.
Davis, Martin E.; Sigal, Ron; Weyuker, Elaine J.. Computability, complexity, and languages:
Fundamentals of theoretical computer science. Boston: Academic Press, Harcourt, Brace. p. 327.
ISBN 0-12-206382-1. External links
http:www.staff.ncl.ac.ukell236/lecture5.htm