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In formal language theory, a context-free language is a language generated by some context-free grammar. The set of all context-free languages is identical to the set of languages accepted by pushdown automata.
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:
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.
Closure properties 
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: .
Decidability properties 
- Equivalence: is ?
- is ? (However, the intersection of a context-free language and a regular language is context-free, so if were a regular language, this problem becomes decidable.)
- is ?
- is ?
The following problems are decidable for arbitrary context-free languages:
- is ?
- is finite?
- Membership: given any word , does ? (membership problem is even polynomially decidable - see CYK algorithm and Earley's Algorithm)
Properties of context-free languages 
- The reverse of a context-free language is context-free, but the complement need not be.
- Every regular language is context-free because it can be described by a context-free grammar.
- The intersection of a context-free language and a regular language is always context-free.
- 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, Parikh's theorem, or using closure properties.
- Context Free Languages are closed under Union, Concatenation, and Kleene star.
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.
See also 
- How to prove that a language is not context-free?
- Properties of Context-Free Languages slides from Stanford
- Knuth, Donald (July 1965). "On the Translation of Languages from Left to Right". Information and Control 8: 707 – 639. Retrieved 29 May 2011.
- Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.
- 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.