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.

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.

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.

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]}

The language of all properly matched parentheses is generated by the grammar .

Main article: 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*(*n*^{2.3728596}).^{[2]}^{[note 2]}
Conversely, Lillian Lee has shown *O*(*n*^{3−ε}) boolean matrix multiplication to be reducible to *O*(*n*^{3−3ε}) 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.

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]}

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]}

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]}

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]}