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.
A model 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.
The language of all properly matched parentheses is generated by the grammar .
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(n2.3728639).[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.
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.
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 circular shift of L (the language )
- the prefix closure of L (the set of all prefixes of strings from L)
- the quotient L/R of L by a regular language R
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: .
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.
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.
- Equivalence: is ?
- Disjointness: is ? However, the intersection of a context-free language and a regular language is context-free, hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
- Containment: is ? Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.
- Universality: 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.
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 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.
- meaning of 's arguments and results:
- In Valiant's papers, O(n2.81) given, the then best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm for bound improvements since then.
- 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.
- Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
- Leslie Valiant (Jan 1974). General context-free recognition in less than cubic time (Technical report). Carnegie Mellon University. p. 11.
- Leslie G. Valiant (1975). "General context-free recognition in less than cubic time". Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8.
- Lillian Lee (2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication" (PDF). J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242.
- Knuth, D. E. (July 1965). "On the translation of languages from left to right" (PDF). Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Retrieved 29 May 2011.
- Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
- Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
- Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
- Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
- Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3: 372–375.
- Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1138. ISBN 1-57955-008-8.
- Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
- Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
- Salomaa (1973), p. 59, Theorem 6.7
- Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
- Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
- Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
- Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
- Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
- Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
- John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411
- Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.
- Hopcroft & Ullman 1979.
- How to prove that a language is not context-free?
- 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.