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In formal language theory, a context-free language (CFL) is a language generated by some context-free grammar (CFG). Different CF grammars can generate the same CF language, or conversely, a given CF language can be generated by different CF grammars. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).
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. Indeed, given a CFG, there is a direct way to produce a pushdown automaton for the grammar (and corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar . Also, most arithmetic expressions are generated by context-free grammars.
An archetypical context-free language is , the language of all non-empty even-length strings, the entire first halves of which are 's, and the entire second halves of which are 's. is generated by the grammar , and is accepted by the pushdown automaton where is defined as follows:
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.
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. 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 or a number of other methods, such as Ogden's lemma or Parikh's theorem.
Context-free languages are 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
- the reversal of L
- the concatenation of L and P
- the Kleene star of L
- the image of L under a homomorphism
- the image of L under an inverse homomorphism
- the cyclic shift of L (the language )
Context-free languages are not closed under complement, intersection, or difference. 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.
Nonclosure under intersection and complement
The context-free languages are not closed under intersection. This can be seen by taking the languages and , which are both context-free. Their intersection is , which can be shown to be non-context-free by the pumping lemma for context-free languages.
Context-free languages are also not closed under complementation, as for any languages A and B: .
- Equivalence: Given two context-free grammars A and B, is ?
- Intersection Emptiness: Given two context-free grammars A and B, is ? However, the intersection of a context-free language and a regular language is context-free, and the variant of the problem where B is a regular grammar is decidable.
- Containment: Given a context-free grammar A, is ? Again, the variant of the problem where B is a regular grammar is decidable.
- Universality: Given a context-free grammar A, is ?
The following problems are decidable for arbitrary context-free languages:
- Emptiness: Given a context-free grammar A, is ?
- Finiteness: Given a context-free grammar A, is finite?
- 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.
Determining an instance of the membership problem; i.e. given a string , determine whether where is the language generated by some grammar ; is also known as parsing.
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 the 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.
See also parsing expression grammar as an alternative approach to grammar and parser.
- Hopcroft & Ullman 1979.
- How to prove that a language is not context-free?
- A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S → Sc | aTb | ε; T → aTb | ε. The grammar for B is analogous.
- Salomaa (1973), p. 59, Theorem 6.7
- Knuth, Donald E. (July 1965). "On the translation of languages from left to right". Information and Control 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Retrieved 29 May 2011.
- Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.
- Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley.
- Arto Salomaa (1973). Formal Languages. ACM Monograph Series.
- Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Chapter 2: Context-Free Languages, pp. 91–122.
- Jean-Michel Autebert, Jean Berstel, Luc Boasson, Context-Free Languages and Push-Down Automata, in: G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag, 1997, 111-174.