In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, is a language generated by a context-free grammar (CFG).
Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.
Background
Context-free grammar
Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.
Automata
The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
Examples
An example context-free language is , the language of all non-empty even-length strings, the entire first halves of which are a's, and the entire second halves of which are b's. L is generated by the grammar
.This language is not regular.It is accepted by the pushdown automaton
where
is defined as follows:[note 1]
Unambiguous CFLs are a proper subset of all CFLs: there are inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of with
. This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset
which is the intersection of these two languages.[1]
Dyck language
The language of all properly matched parentheses is generated by the grammar .
Properties
Context-free parsing
The context-free nature of the language makes it simple to parse with a pushdown automaton.
Determining an instance of the membership problem; i.e. given a string , determine whether
where
is the language generated by a given grammar
; is also known as recognition. Context-free recognition for Chomsky normal form grammars was shown by Leslie G. Valiant to be reducible to boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728596).[2][note 2]Conversely, Lillian Lee has shown O(n3−ε) boolean matrix multiplication to be reducible to O(n3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.[3]
Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called parsing. Known parsers have a time complexity that is cubic in the size of the string that is parsed.
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the CYK algorithm and Earley's Algorithm.
A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR(k) parser.[4]
See also parsing expression grammar as an alternative approach to grammar and parser.
Closure properties
The class of context-free languages is closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:
- the union
of L and P[5]
- the reversal of L[6]
- the concatenation
of L and P[5]
- the Kleene star
of L[5]
- the image
of L under a homomorphism
[7]
- the image
of L under an inverse homomorphism
[8]
- the circular shift of L (the language
)[9]
- the prefix closure of L (the set of all prefixes of strings from L)[10]
- the quotient L/R of L by a regular language R[11]
Nonclosure under intersection, complement, and difference
The context-free languages are not closed under intersection. This can be seen by taking the languages and
, which are both context-free.[note 3] Their intersection is
, which can be shown to be non-context-free by the pumping lemma for context-free languages. As a consequence, context-free languages cannot be closed under complementation, as for any languages A and B, their intersection can be expressed by union and complement:
. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference:
.[12]
However, if L is a context-free language and D is a regular language then both their intersection and their difference
are context-free languages.[13]
Decidability
In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.
The following problems are undecidable for arbitrarily given context-free grammars A and B:
- Equivalence: is
?[14]
- Disjointness: is
?[15] However, the intersection of a context-free language and a regular language is context-free,[16][17] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
- Containment: is
?[18] Again, the variant of the problem where B is a regular grammar is decidable,[citation needed] while that where A is regular is generally not.[19]
- Universality: is
?[20]
- Regularity: is
a regular language?[21]
- Ambiguity: is every grammar for
ambiguous?[22]
The following problems are decidable for arbitrary context-free languages:
- Emptiness: Given a context-free grammar A, is
?[23]
- Finiteness: Given a context-free grammar A, is
finite?[24]
- Membership: Given a context-free grammar G, and a word
, does
? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.
According to Hopcroft, Motwani, Ullman (2003),[25] many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of Bar-Hillel, Perles, and Shamir[26]
Languages that are not context-free
The set is a context-sensitive language, but there does not exist a context-free grammar generating this language.[27] So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages[26] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[28]
Notes
References
Works cited
- Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 9780201029888.
- Salomaa, Arto (1973). Formal Languages. ACM Monograph Series.
Further reading
- Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata". In G. Rozenberg; A. Salomaa (eds.). Handbook of Formal Languages (PDF). Vol. 1. Springer-Verlag. pp. 111–174. Archived (PDF) from the original on 2011-05-16.
- Ginsburg, Seymour (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill.
- Sipser, Michael (1997). "2: Context-Free Languages". Introduction to the Theory of Computation. PWS Publishing. pp. 91–122. ISBN 0-534-94728-X.