Context-free language
In formal language theory, a context-free language (CFL) 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 formula_1, 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 formula_2. This language is not regular. It is accepted by the pushdown automaton formula_3 where formula_4 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 formula_6 with formula_7. 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 formula_8 which is the intersection of these two languages. Dyck language. The language of all properly matched parentheses is generated by the grammar formula_9. 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 formula_10, determine whether formula_11 where formula_12 is the language generated by a given grammar formula_13; 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"("n"). Conversely, Lillian Lee has shown "O"("n") boolean matrix multiplication to be reducible to "O"("n") CFG parsing, thus establishing some kind of lower bound for the latter. 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. See also parsing expression grammar as an alternative approach to grammar and parser. Closure. 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: Nonclosure under intersection, complement, and difference. The context-free languages are not closed under intersection. This can be seen by taking the languages formula_22 and formula_23, which are both context-free. Their intersection is formula_24, 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: formula_25. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: formula_26. However, if "L" is a context-free language and "D" is a regular language then both their intersection formula_27 and their difference formula_28 are context-free languages. 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: The following problems are "decidable" for arbitrary context-free languages: According to Hopcroft, Motwani, Ullman (2003), 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 Languages that are not context-free. The set formula_8  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.